Irresolute Homotopy and Covering Theory in Irresolute Topological Groups ^†

Abstract

In this paper, we explore certain properties related to connectedness and introduce the definition of irresolute paths. Subsequently, we define the concepts of semi-path connectedness, locally semi-path connectedness, and semi-locally s-simply connected spaces. Additionally, we introduce the concept of irresolute homotopy and reconstruct the fundamental group based on this framework. Furthermore, we prove that the structure of an irresolute topological group with a universal irresolute covering can be lifted to its irresolute covering space.

1. Introduction

Levine [1] first defined the concept of semi-open sets in 1963 and investigated their properties in topological spaces. Many new studies have been conducted using the notions of semi-open sets and semi-continuity [2,3,4,5]. Later, in 1996, Jordan used the concept of semi-open sets to explore widely known topological notions such as the semi-closure, semi-interior, semi-boundary, and semi-exterior of a set. In 1974, Das [6] introduced the concept of semi-connectedness as a weaker variant of connectedness in topological spaces. The exploration of connectedness and its various generalizations remains a significant aspect of topology. The concept of a locally semi-connected set is given in [7]. Semi*-open and semi*-closed sets were introduced in [8].

In [9], irresolute maps and related separation axioms were introduced, and various irresolute transformation groups were defined.

Kočinac introduced s-topological and S-topological groups using semi-open sets [10]. In later studies, irresolute topological groups were defined and their properties were examined using irresolute functions [11]. A new type of topological ring, called an irresolute topological ring (or semitopological ring), was introduced in [12].

A path in a space T is defined as a continuous map $f:I\to T$, where I is the unit interval $[0,1]$. The concept of continuously deforming a path while keeping its endpoints fixed is captured by the notion of homotopy. Two paths with the same endpoints are said to be homotopic if one can be continuously transformed into the other while preserving the endpoints and staying within the defined region [13].

In [14], it was shown that irresolute topological vector spaces possess several important properties: they are open-hereditary; homomorphisms are irresolute if and only if they are irresolute at the identity; scalar multiples of semi-compact sets remain semi-compact; and every semi-open set is translationally invariant.

The fundamental group of a topological space is the group of equivalence classes of loops under homotopy in that space [13].

The study of covering spaces holds a fundamental place in algebraic topology. A well-established result states that, if T is a topological group with an additive structure and $c:\tilde{T}\to T$ is a simply connected covering map, then given a point $\tilde{0}\in \tilde{T}$ such that $c\left(\tilde{0}\right)=0$, the space $\tilde{T}$ inherits a topological group structure with identity element $\tilde{0}$, and c becomes a homomorphism between topological groups (see, for instance, [15]). Furthermore, a topological bounded cohomology theory for topological semigroups was established in [16].

The challenge of identifying universal covers for topological groups that are not connected was initially addressed in [17]. In this work, Taylor demonstrated that a topological group T gives rise to an obstruction class $k_T$ in the cohomology group $H^3({\pi }_0\left(T\right);{\pi }_1(T,0))$. The vanishing of this class is necessary and sufficient for the lifting of the group structure to a universal covering space.

A related algebraic characterization was later presented in [18] using crossed modules and group-groupoids, which are group objects within the category of groupoids. Additionally, a revised and more general version of this result can be found in [19], where the connection to obstruction theory for group extensions is established. Furthermore, the concept of monodromy in topological group-groupoids, introduced in [20], provides a modern perspective on these structures. In [21], coverings of groups with operations were studied. Also, in [22,23], the lifts of multi-operation topological groups and the lifts of local topological group structures, respectively, were presented.

In this paper, the results previously established for local topological groups, topological groups, and their generalizations such as groups with operations are reconstructed in the framework of irresolute topological groups, as introduced in [11].

In Section 2, we present the basic definitions and known results related to semi-open sets, semi-continuity, and irresolute functions. In Section 3, we define irresolute paths and semi-path connectedness, and examine their relationship with semi-connectedness. We also introduce the concept of irresolute homotopy and construct the irresolute fundamental group. In Section 4, we discuss the notion of irresolute covering maps and explore their properties. Finally, in Section 5, we investigate the lifting of algebraic operations to irresolute covering spaces of irresolute topological groups.

2. Preliminaries

In a topological space T, a subset S is called semi-open if there is an open set U in T such that $U\subset S\subset \mathrm{Cl}\left(U\right)$, or equivalently, if $S\subset \mathrm{Cl}\left(\mathrm{Int}\right(S\left)\right)$ [1]. The collection of all semi-open sets in T is denoted by $SO\left(T\right)$.

Definition 1 

([3]). Let $(T,\tau )$ be a topological space and S be a subset of T. Then $p\in T$ is called a semi-interior point of S if there is a semi-open set U such that $p\in U\subset S$ (or each semi-open neighborhood of p meets S).

The set of all semi-interior points of S is called the semi-interior of S, denoted $sInt\left(A\right)$.

The complement of a semi-open set is known as a semi-closed set. The semi-closure of a subset S, written as $\mathit{sCl}\left(S\right)$, is the intersection of all semi-closed sets that contain S [3,4].

A point p belongs to $\mathit{sCl}\left(S\right)$ if and only if every semi-open set that contains p also intersects S, that is, $U\cap S\ne \varnothing$ for any semi-open set U containing p.

Every open (or closed) set is also semi-open (or semi-closed). It is well established that the union of any number of semi-open sets remains semi-open. However, the intersection of two semi-open sets is not necessarily semi-open. Additionally, the intersection of an open set with a semi-open set always results in a semi-open set. Furthermore, if $S\subset T$ and $O\subset Y$ are semi-open in their respective spaces T and Y, then their Cartesian product $S\times O$ is also semi-open in the product space $T\times Y$.

Definition 2. 

A mapping $f:T\to Y$ between topological spaces T and Y is labeled as follows:

(i) 

Semi-continuous  [1] (or irresolute [4]) if for each open (or semi-open) set $V\subset Y$, the set $f^{-1}\left(V\right)$ is semi-open in T. Evidently, the mapping f is semi-open (irresolute) if, for each $p\in T$, and for each open (semi-open) neighborhood V of $f\left(p\right)$, there is a semi-open neighborhood U of p such that $f\left(U\right)\subset V$.

(ii) 

(Ref. [4]) Pre-semi-open  if, for every semi-open set S of T, $f\left(S\right)$ is semi-open in Y.

(iii) 

(Ref. [4]) A semi-homeomorphism  if f is bijective, irresolute, and pre-semi-open.

Definition 3. 

Two non-empty subsets $S,O$ of a topological space $(T,\tau )$ are said to be semi-separated  [6], if and only if $S\cap \mathit{sCl}\left(O\right)=\mathit{sCl}\left(S\right)\cap O=\varnothing$.

Definition 4 

([6]). In a topological space $(T,\tau )$, a set that cannot be expressed as the union of two semi-separated sets is called a semi-connected set. The topological space $(T,\tau )$ is said to be semi-connected if and only if T itself is semi-connected.

Lemma 1 

([1]). Let $S\subset Y\subset T$, where T is a topological space and Y is a subspace. If $S\in \mathit{SO}\left(T\right)$, then $S\in \mathit{SO}\left(Y\right)$.

Lemma 2 

([7]). Let $f:(T,\tau )\to (Y,\sigma )$ be an open and semi-continuous function, and let $S\subset T$ be open. Then $f\left(S\right)$ is semi-connected, provided that S is semi-connected.

Lemma 3 

([6]). The semi-homeomorphic image of a semi-connected space is semi-connected.

Proof. 

Let T be a semi-connected space, and let $f:T\to Y$ be a semi-homeomorphism. Suppose, for contradiction, that $Y=f\left(T\right)$ is semi-disconnected. Then, there are two non-empty semi-separated open sets S and o in Y such that

$$S\cup B=f\left(T\right).$$

Since $f^{-1}$ is irresolute, the preimages $f^{-1}\left(S\right)$ and $f^{-1}\left(O\right)$ are semi-open in T. Moreover, since f is a bijection, $f^{-1}\left(S\right)$ and $f^{-1}\left(O\right)$ are non-empty and semi-separated, and their union is T. This contradicts the assumption that T is semi-connected. □

Definition 5 

([11]). A topologized group $(G,\ast ,\tau )$ is called an irresolute-topological group if, for each $p,y\in G$ and each semi-open neighborhood R of $p\ast y^{-1}$ in G, there are semi-open neighborhoods P of p and O of y such that

$$P\ast O^{-1}\subset R.$$

Theorem 1 

([11]). If $(G,\ast ,\tau )$ is an irresolute-topological group, then the multiplication mapping

$$m:G\times G\to G$$

and the inverse mapping

$$i:G\to G$$

are irresolute.

3. Irresolute Path Connectedness

In this section, some properties related to connectedness and the definition of irresolute paths will be provided. Subsequently, the concepts of semi-path connected, locally semi-path connected, and semi-locally s-simply connected will be introduced. Furthermore, the concept of irresolute homotopy is introduced, and the fundamental group is reconstructed. Before presenting these definitions, let us prove the pasting lemma for irresolute functions.

Theorem 2. 

Let $\left\{S_j\mid j\in J\right\}$ be a collection of semi-open subsets of a topological space T, and let $f_j:S_j\to Y$ be irresolute maps such that for all $p\in S_k\cap S_j$, we have

$$f_j\left(p\right)=f_k\left(p\right).$$

Let $S={\bigcup }_{j\in I}{S}_j$. Then there is a unique map $f:S\to Y$ such that

$${f|}_{S_j}=f_j\mathit{for}\mathit{all}j\in J.$$

Proof. 

Since each $S_j$ is semi-open in T, it is also semi-open in S.

We first define a function $f:S\to Y$ as follows: For any $p\in S$, pick any $j\in J$ such that $p\in S_j$ and define

$$f\left(p\right)=f_j\left(p\right).$$

Such an index j exists because S is the union of the $S_j$’s. Moreover, the definition of $f\left(p\right)$ is independent of the choice of i because, if $p\in S_k\cap S_j$, then

$$f_k\left(p\right)=f_j\left(p\right).$$

Thus, the function f is well-defined and is the only possible way to define such a map.

To show that f is irresolute, let S be a semi-open subset of Y. We need to show that $f^{-1}\left(S\right)$ is semi-open in S.

If $f\left(p\right)\in S$, then $f_j\left(p\right)\in S$ for some j. Thus,

$$f^{-1}\left(S\right)=\bigcup _{j\in J}{f}_j^{-1}\left(S\right).$$

Since each $f_j:S_j\to Y$ is irresolute, $f_j^{-1}\left(S\right)$ is semi-open in $S_j$. Since semi-open subsets of semi-open subsets are open, and each $S_j$ is semi-open in S, it follows that $f_j^{-1}\left(S\right)$ is open in S. Therefore, the union ${\bigcup }_{j\in J}{f}_j^{-1}\left(S\right)$ is also a semi-open subset of S, proving that f is irresolute. □

Remark 1. 

We recall that the classical pasting lemma for continuous functions holds when the domain is covered by closed sets and the functions agree on the overlaps. In the case of irresolute functions, a similar result remains valid when the domain is covered by semi-closed sets. That is, if a space is covered by semi-closed subsets and the function is irresolute on each subset with consistent behavior on their intersection, then the combined function is also irresolute.

Definition 6. 

Let $\xi :[0,1]\to T$ be an irresolute function. If $\xi \left(0\right)=p$ and $\xi \left(1\right)=r$, then the path ξ is called an irresolute path  from p to r.

Definition 7. 

A space T is irresolute path-connected  if for all points $p,r\in T$, there is a path from p to r, that is, an irresolute map

$$\xi :[0,1]\to T$$

such that

$$\xi \left(0\right)=p\mathit{and}\xi \left(1\right)=r.$$

Example 1. 

Let $T=\{p,r\}$ and $\tau =\{T,\varnothing ,\{p\left\}\right\}$ be a topology on T. Then the topological space $(T,\tau )$ is not an irresolute path-connected space which is a path-connected space.

Example 2. 

A set T is neither path-connected nor irresolute path-connected under the discrete topology. However, under the indiscrete topology, it is both path-connected and irresolute path-connected. It should be noted that path-connectedness and irresolute path-connectedness are independent of each other.

Theorem 3. 

Let T be a topological space. The relation on T given by “$p\sim r$ if there is an irresolute path in T connecting p and r" defines an equivalence relation.

Proof. 

We establish that this relation satisfies the properties of an equivalence relation:

Reflexivity: For each $p\in T$, consider a constant function $f:[0,1]\to T$ defined by

$$f\left(t\right)=p,\forall t\in [0,1].$$

Since the function f takes the constant value p at each point $t\in [0,1]$, it is a valid irresolute path from p to itself.

Symmetry: Given an irresolute path $f:[0,1]\to T$ from p to r, we define a new function $g:[0,1]\to T$ by

$$g\left(t\right)=f(1-t).$$

This function traverses the irresolute path in reverse, ensuring a connection from r back to p.

Transitivity: Suppose $f:[0,1]\to T$ is a path joining p to r and $g:[0,1]\to T$ is an irresolute path linking r to w. We define an irresolute function $h:[0,1]\to T$ as follows:

$$h\left(t\right)=\left\{\begin{array}{cc}f\left(2t\right),\hfill & 0\le t\le {\displaystyle \frac{1}{2}},\hfill \\ g(2t-1),\hfill & {\displaystyle \frac{1}{2}}\le t\le 1.\hfill \end{array}\right.$$

Note that the definition of the concatenated path function at $t=0.5$ is valid due to Theorem 2 for irresolute functions defined on semi-closed subsets. For further clarification, see Remark 1.

Since the relation satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation. □

Definition 8. 

The equivalence classes of T under the relation defined in Theorem 3 are referred to as the irresolute path components  of T.

It follows that any topological space can be expressed as the disjoint union of its irresolute path-connected subspaces, which are precisely its irresolute path components.

Before presenting the relationship between irresolute path- connectedness and semi-connectedness, let us state the following proposition.

Proposition 1. 

If M is a proper subset of $\mathbb{R}$ that is semi-connected in the usual topology of $\mathbb{R}$, then M must be an interval.

Proof. 

Assume, for contradiction, that A is not an interval. Choose a point $p\in \mathbb{R}\setminus A$. Define the sets

$$P=M\cap (-\infty ,p)\mathrm{and}V=M\cap (p,\infty ).$$

Since $p\notin M$, both P and V are non-empty, and they satisfy

$$M=P\cup V,sCl\left(P\right)\cap V=\varnothing \mathrm{and}P\cap scl\left(V\right)=\varnothing .$$

Moreover, as P and V are semi-open in M, it follows that M is semi-disconnected, contradicting the assumption that M is semi-connected. Hence, M must be an interval. □

Theorem 4 

(Intermediate value theorem). Let T be a semi-connected topological space, and let $f:T\to Y$ be an irresolute function. If $p,r\in f\left(T\right)$, then f attains all values between p and r.

Proof. 

Since T is semi-connected, it follows from Theorem 3 that $f\left(T\right)$ is semi-connected. If f is onto, the proof is trivial. If f is not onto, then by Proposition 1, $f\left(T\right)$ is an interval. Hence, for any interval $[p,r]\subseteq f\left(T\right)$ with $p<r$, there is some $p\in T$ such that $f\left(p\right)=s$ for every $s\in [p,r]$. □

Proposition 2. 

Any interval in the usual space of $\mathbb{R}$ is semi-connected.

Proof. 

It is sufficient to prove the claim for the closed interval $[p,r]$, as the other cases can be shown similarly. Suppose that the set $M=[p,r]$ is semi-disconnected. Then, there are open subsets $P,V\subset \mathbb{R}$ such that

$$(M\cap P)\cup (M\cap V)=M\mathrm{and}(P\cap V)\cap A=\varnothing .$$

Note that, since $M\cap P$ and $M\cap V$ are open, then semi-open. Define the function $f:[p,r]\to \mathbb{R}$ as follows:

$$f^{-1}\left(p\right)=\left\{\begin{array}{cc}1,\hfill & p\in M\cap P,\hfill \\ 0,\hfill & p\in M\cap V.\hfill \end{array}\right.$$

The function f is irresolute because, for any semi-open set $U\subset \mathbb{R}$, the preimage $f^{-1}\left(U\right)$ is given by

$$f^{-1}\left(U\right)=\left\{\begin{array}{cc}M\cap P,\hfill & 1\in U,0\notin U,\hfill \\ M\cap V,\hfill & 0\in U,1\notin U,\hfill \\ M,\hfill & 0,1\in U,\hfill \\ \varnothing ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In each case, $f^{-1}\left(U\right)$ is semi-open in M, confirming that f is irresolute. Consequently, from Theorem 4 f must take all values in the interval $[0,1]$. However, this leads to a contradiction since there is no $m\in M$ such that $f\left(m\right)={\displaystyle \frac{1}{2}}$. Hence, $M=[p,r]$ must be semi-connected. □

Definition 9. 

A topological space is called locally irresolute path-connected if, for every point p and neighbourhood $P\ni p$, there is a path-connected neighbourhood $U\subset P$ that contains p.

Example 3. 

The space $X=\{p,r,s\}$ with the topology

$$\tau =\{X,\varnothing ,\{p\},\{r\},\{p,r\left\}\right\}$$

is not irresolute path-connected but is locally irresolute path-connected.

Theorem 5. 

A locally irresolute path-connected topological space is semi connected if and only if it is irresolute path-connected.

Proof. 

Since T is semi-connected, T has only one semi-component; since T is locally irresolute path-connected, this component is an irresolute path component. □

Let us define the irresolute homotopy of functions. We will then define the irresolute homotopy of paths and construct the equivalence classes necessary for the fundamental group.

Definition 10. 

Let $f,g:T\to Y$ be irresolute functions. An irresolute function

$\Sigma :T\times [0,1]\to Y$ satisfying

$$\Sigma (p,0)=f\left(p\right)$$

$$\Sigma (p,1)=g\left(p\right)$$

is called an irresolute homotopy  , and the functions f and g are said to be irresolutely homotopic  . This is denoted as $f\simeq g$ or $F:f\simeq g$.

For an irresolute homotopy Σ, we define ${\Sigma }_0=f$ and ${\Sigma }_1=g$ such that, for every $p\in [0,1]$, the function

$${\Sigma }_t\left(x\right)=\Sigma (p,t)$$

is an irresolute function ${\Sigma }_t:T\to Y$.

Thus, such a homotopy can be written as a class of irresolute functions $\{{\Sigma }_t:0\le t\le 1\}$. Therefore, if the functions f and g can be irresolutely deformed into each other through the same type of irresolute functions, then they are irresolutely homotopic.

Example 4. 

Let $f,g:S^1\to S^1$ be irresolute functions. Define:

$$f\left(p\right)=p,g\left(p\right)=p^2$$

Both f and g are irresolute.

Indeed, since f is the identity function, the preimage of every semi-open set is also semi-open, so f is irresolute.

For the function g, consider the following cases:

$$P_1=(1,4),g^{-1}\left(P_1\right)=(-2,-1)\cup (1,2)$$

$$P_2=[1,4),g^{-1}\left(P_2\right)=(-2,-1]\cup [1,2)$$

$$P_3=(1,4],g^{-1}\left(P_3\right)=[-2,-1)\cup (1,2]$$

Since the preimage of every semi-open set under g is also semi-open, it follows that g is irresolute.

Furthermore, f and g are irresolutely homotopic. Indeed, the function

$F:S^1\times [0,1]\to S^1$ maps the preimage of every semi-open set to a semi-open set. Thus, F is an irresolute homotopy between f and g.

Definition 11. 

Let $\lambda ,\nu :[0,1]\to T$ be two irresolute paths from p to r in a topological space T. If there is an irresolute function

$$\Sigma :[0,1]\times [0,1]\to T$$

such that, for $0\le s\le 1$ and $0\le t\le 1$,

$$\Sigma (s,0)=\lambda \left(s\right),$$

$$\Sigma (s,1)=\nu \left(s\right),$$

$$\Sigma (0,t)=p,$$

$$\Sigma (1,t)=r,$$

then λ and ν are called irresolutely homotopic with respect to the endpoints, denoted by $\lambda {\simeq }_{\mathrm{ir}}\nu$. Such a Σ is called an irresolute homotopy and is denoted by $\Sigma :\lambda {\simeq }_{\mathrm{ir}}\nu$.

In this irresolute homotopy function, for ${\Sigma }_0=\lambda$ and ${\Sigma }_1=\nu$, ${\Sigma }_t\left(s\right)=\Sigma (s,t)$ is an irresolute path for all $t\in [0,1]$. Therefore, such an irresolute homotopy is usually written as

$$\{{\Sigma }_t:0\le t\le 1\}.$$

Proposition 3. 

If $\lambda {\simeq }_{\mathrm{ir}}{\lambda }^{\prime }$ and $\nu {\simeq }_{\mathrm{ir}}{\nu }^{\prime }$, then $\lambda \diamond \nu {\simeq }_{\mathrm{ir}}{\lambda }^{\prime }\diamond {\nu }^{\prime }$.

Definition 12. 

Let T be a topological space and $p_0\in T$. An irresolute loop  based at $p_0$ is an irresolute path $\lambda :I\to T$ such that $\lambda \left(0\right)=\lambda \left(1\right)=p_0$. Two such loops λ and ν are called irresolutely homotopic  (denoted $\lambda {\simeq }_{\mathrm{ir}}\nu$) if there is an irresolute homotopy between them that preserves endpoints.

Definition 13. 

The set of all irresolute homotopy classes of loops based at $p_0$ is denoted by ${\pi }_1^{\mathrm{ir}}(T,p_0)$ and is called the irresolute fundamental group of T at $p_0$.

Theorem 6. 

Let T be a semi-connected, locally irresolute path-connected, and semi-locally s-simply connected topological space. Then ${\pi }_1^{\mathrm{ir}}(T,p_0)$, the set of irresolute homotopy classes of loops based at $p_0$, forms a group under the operation of concatenation of paths.

Let T be a topological space and $p_0\in T$. An irresolute loop based at $p_0$ is defined as an irresolute path $\lambda :I\to T$ such that $\lambda \left(0\right)=\lambda \left(1\right)=p_0$. Two such loops $\lambda$ and $\nu$ are said to be irresolutely homotopic, written $\lambda {\simeq }_{\mathrm{ir}}\nu$, if there is an irresolute homotopy $\Sigma :I\times I\to T$ such that

$$\Sigma (s,0)=\lambda \left(s\right),\Sigma (s,1)=\nu \left(s\right),\Sigma (0,t)=\Sigma (1,t)=p_0$$

for all $s,t\in [0,1]$. The collection of all such homotopy classes forms a set, which we denote by ${\pi }_1^{\mathrm{ir}}(T,p_0)$ and call the irresolute fundamental group of T at the base point $p_0$.

Theorem 7. 

Let T be a semi-connected, locally irresolute path-connected, and semi-locally s-simply connected topological space. Then ${\pi }_1^{\mathrm{ir}}(T,p_0)$, the set of irresolute homotopy classes of closed paths based at $p_0$, forms a group under the operation of path concatenation.

Proof. 

Let $\lambda ,\nu \in {\pi }_1^{\mathrm{ir}}(T,p_0)$ be two irresolute loops based at $p_0$. Define the followin operation:

$$(\lambda \diamond \nu )\left(t\right)=\left\{\begin{array}{cc}\lambda \left(2t\right),\hfill & \mathrm{if}0\le t\le \frac{1}{2},\hfill \\ \nu (2t-1),\hfill & \mathrm{if}\frac{1}{2}\le t\le 1.\hfill \end{array}\right.$$

By Theorem 2 (since $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ are semi-closed), $\lambda \diamond \nu$ is an irresolute path. The associativity, identity (trivial loop), and inverses under concatenation follow similarly as in the classical case, with each operation preserving irresoluteness due to closure under composition. Thus, ${\pi }_1^{\mathrm{ir}}(T,p_0)$ forms a group. □

Theorem 8. 

Let T be an irresolute topological group, and let $p_0\in T$. The set of irresolute homotopy classes of all closed paths in T from $p_0$ to $p_0$, denoted by ${\pi }_1^{\mathrm{ir}}(T,p_0)$, forms a group.

Proof. 

Let T be an irresolute topological group and let $P(T,p_0)$ denote the set of all irresolute paths in T that start at $p_0$. We define binary operations on $P(T,p_0)$ by

$$(\lambda \ast \nu )\left(s\right)=\lambda \left(s\right)\ast \nu \left(s\right),$$

for $s\in J$, where J is the unit interval. Additionally, we introduce inverse operations given by

$$\left({\lambda }^{-1}\right)\left(s\right)={\left(\lambda \left(s\right)\right)}^{-1}.$$

Consequently, these operations induce binary operations on the fundamental group ${\pi }_1^{\mathrm{ir}}(T,p_0)$ as follows:

$$\left[\lambda \right]\ast \left[\nu \right]=[\lambda \ast \nu ],$$

(1)

for any irresolute homotopy classes $\left[\lambda \right],\left[\nu \right]\in {\pi }_1^{\mathrm{ir}}(T,p_0)$. From Proposition 3, since the binary operation * is irresolute, it ensures that the induced binary operation is well-defined. Similarly, the unary operations lead to the unary operations on ${\pi }_1^{\mathrm{ir}}(T,p_0)$:

$${\left[\lambda \right]}^{-1}=\left[{\lambda }^{-1}\right].$$

(2)

Since the unary operation is irresolute, it follows that the operation is also well-defined. By verifying additional details, we conclude that ${\pi }_1^{\mathrm{ir}}(T,p_0)$ forms a group. □

Definition 14. 

Let $h:(T,p_0)\to (Y,y_0)$ be an irresolute function. Then we define the map

$$h_{*}:{\pi }_1^{\mathrm{ir}}(T,p_0)\to {\pi }_1^{\mathrm{ir}}(Y,y_0),$$

which is known as the homomorphism induced by  h, relative to the base point $p_0$, and is given by

$$h_{*}\left(\left[f\right]\right)=[h\circ f].$$

Definition 15. 

A topological space T is semilocally s-simply connected if, for every point $p\in T$, there is a semi-neighborhood P of p such that the map of fundamental groups

$${\pi }_1^{\mathrm{ir}}(P,p)\to {\pi }_1^{\mathrm{ir}}(T,p)$$

induced by the inclusion map $P\hookrightarrow T$ is the trivial homomorphism.

Corollary 1. 

If

$$h:T\to S$$

is a semi-homeomorphism, then the fundamental groups ${\pi }_1^{\mathrm{ir}}(T,p_0)$ and ${\pi }_1^{\mathrm{ir}}(S,h\left(p_0\right))$ are isomorphic, i.e.,

$${\pi }_1^{\mathrm{ir}}(T,p_0)\cong {\pi }_1^{\mathrm{ir}}(S,h\left(p_0\right)).$$

Theorem 9. 

Let T be a topological space and $p,q\in T$. If there is an irresolute path from p to q, then the fundamental groups ${\pi }_1^{\mathrm{ir}}(T,p)$ and ${\pi }_1^{\mathrm{ir}}(T,q)$ are isomorphic.

Proof. 

Let $\lambda$ be an irresolute path from p to r. A group isomorphism from ${\pi }_1^{\mathrm{ir}}(T,p)$ to ${\pi }_1^{\mathrm{ir}}(T,r)$ is defined by

$${\lambda }_{*}:{\pi }_1^{\mathrm{ir}}(T,p)\to {\pi }_1^{\mathrm{ir}}(T,r),\left[\zeta \right]\mapsto {\left[\lambda \right]}^{-1}\left[\zeta \right]\left[\lambda \right].$$

Verifying that ${\lambda }_{*}$ is a group isomorphism is straightforward. Note that the inverse of ${\lambda }_{*}$ is given by

$${\lambda }_{\ast }^{-1}:{\pi }_1^{\mathrm{ir}}(T,r)\to {\pi }_1^{\mathrm{ir}}(T,p),\left[\zeta \right]\mapsto \left[\lambda \right]\left[\zeta \right]{\left[\lambda \right]}^{-1}.$$

□

Corollary 2. 

In an irresolute path-connected topological space, all fundamental groups are isomorphic to each other.

Let T be a topological group with operations. By evaluating the compositions and operations of the paths in T such that ${\lambda }_1\left(1\right)={\zeta }_1\left(0\right)$ and ${\lambda }_2\left(1\right)={\zeta }_2\left(0\right)$, we establish the following interchange law:

$$({\lambda }_1\circ {\zeta }_1)\ast ({\lambda }_2\circ {\zeta }_2)=({\lambda }_1\ast {\lambda }_2)\circ ({\zeta }_1\ast {\zeta }_2)$$

(3)

for $\ast \in {\Omega }_2$, where ∘ represents the composition of paths, and

$$\overline{(\lambda \ast \zeta )}=\overline{\lambda }\ast \overline{\zeta }$$

(4)

for $\lambda ,\zeta \in P(T,0)$, where the inverse path $\overline{\lambda }$ is given by

$$\overline{\lambda }\left(p\right)=\lambda (1-p)\mathrm{for}p\in I.$$

Additionally, we obtain

$${\left(\overline{\lambda }\right)}^{-1}=\overline{{\lambda }^{-1}},$$

(5)

$${(\lambda \circ \zeta )}^{-1}=\left({\lambda }^{-1}\right)\circ \left({\zeta }^{-1}\right)$$

(6)

when $\lambda \left(1\right)=\zeta \left(0\right)$.

4. Irresolute Coverings of Topological Spaces

In this section, we will define the coverings of topological spaces using irresolute functions and provide some of their properties.

Definition 16. 

Let $\tilde{T}$ and T be semi-connected and locally irresolute path-connected topological spaces. Let $c_i:\tilde{T}\to T$ be an irresolute function.

For a semi-open set $P\subseteq T$, the preimage $c_i^{-1}\left(P\right)$ is said to be irresolute-covered by $c_i$or simply irresolute-canonical  if each of its irresolute path-connected components is irresolute-homeomorphic to P.

If, for every point $p\in T$, there is an irresolute canonical open set P, then $c_i:\tilde{T}\to T$ is called an irresolute-covering function  , and $\tilde{T}$ is referred to as the irresolute-covering space  of T.

Example 5. 

For every topological space T, the identity map

$$id:T\to T$$

is a covering map.

Example 6. 

The function known as the covering map,

$$c_i:\mathbb{R}\to S^1,c_i\left(p\right)=e^{2\pi ip}$$

is an irresolute covering function. Hence, it is an irresolute-covering because the sets

$$P_1=S^1\setminus \left\{1\right\},P_2=S^1\setminus \{-1\}$$

form a semi-open cover of $S^1$. Here,

$$c_i^{-1}\left(P_1\right)=\bigcup _{n\in \mathbb{N}}(n,n+1),$$

$$c_i^{-1}\left(P_2\right)=\bigcup _{n\in \mathbb{N}}\left(n-{\displaystyle \frac{1}{2}},n+{\displaystyle \frac{1}{2}}\right)$$

where each irresolute path-connected component of $c_i^{-1}\left(P_1\right)$ is semi-homeomorphic to $P_1$, and each irresolute path-connected component of $c_i^{-1}\left(P_2\right)$ is semi-homeomorphic to $P_2$. Thus, $P_1$ and $R_2$ are irresolute canonical sets.

Definition 17. 

Let $c_i:\tilde{T}\to T$ be an irresolute-covering map, and let P be an arbitrary topological space. An irresolute function $\tilde{f}:P\to \tilde{T}$ is called the irresolute-lift of f, if $c_i\circ \tilde{f}=f$.

If the function $c_i$ can be lifted to every covering of T, then $c_i$ is called the irresolute universal covering.

Remark 2. 

Let T be a topological space that is semi-connected, locally irresolute path-connected, and semi-locally s-simply connected. Consider an irresolute covering map

$$q:(\tilde{T},\tilde{p})\to (T,p_0).$$

Denote the characteristic subgroup associated with q as G. Then, the irresolute-covering map q is equivalent to the irresolute-covering map

$$c_i:({\tilde{T}}_G,{\tilde{p}}_0)\to (T,p_0)$$

that corresponds to the subgroup G.

Theorem 10. 

Let $(T,p_0)$ be a pointed topological space, and let G be a subgroup of ${\pi }_1^{\mathrm{ir}}(T,p_0)$. If T is semi-connected, locally irresolute path-connected, and semi-locally s-simply connected, then there is an irresolute-covering space $({\tilde{T}}_G,c_i)$ such that $c_{i_{*}}{\pi }_1^{\mathrm{ir}}({\tilde{T}}_G,{\tilde{p}}_0)=G$.

Proof. 

Since T is semi-locally s-simply connected, for each point $p\in T$, there is a semi-open neighborhood R of p such that every irresolute loop in R based at p is irresolute nullhomotopic in T. Additionally, since T is locally path-connected, we can find an open, irresolute path-connected neighborhood P of p such that $p\in P\subset R$. This ensures that every irresolute loop in P based at p is also irresolute nullhomotopic in T.

To show that $({\tilde{T}}_G,c_i)$ is an irresolute-covering space of T, consider a point $\tilde{p}\in c_i^{-1}\left(p\right)$ such that $\tilde{p}={\left[\theta \right]}_G$, where $\theta$ is an irresolute path in T from $p_0$ to p. We claim that $(P,\tilde{p})$ forms a sheet over P in the covering space.

First, we show that $c_i{|}_{(P,\tilde{p})}:(P,\tilde{p})\to P$ is surjective. Given $s\in P$, since P is irreoslute path-connected, there is an irresolute path $\beta$ in P from p to s. Then, the irresolute path $\theta \ast \beta$ (the concatenation of $\theta$ and $\beta$) is a continuation of $\theta$ within P such that $(\theta \ast \beta )\left(1\right)=s$, implying that ${[\theta \ast \beta ]}_G\in (P,\tilde{p})$ and $c_i\left({[\theta \ast \beta ]}_G\right)=s$.

Next, we show that $c_i{|}_{(P,\tilde{p})}$ is injective. Suppose ${\tilde{s}}_1,{\tilde{s}}_2\in (P,\tilde{p})$ satisfy $c_i\left({\tilde{s}}_1\right)=c_i\left({\tilde{s}}_2\right)=s$. Then there are irresolute paths ${\beta }_1$ and ${\beta }_2$ in P such that ${\tilde{s}}_1={[\theta \ast {\beta }_1]}_G$ and ${\tilde{s}}_2={[\theta \ast {\beta }_2]}_G$. Since P is chosen such that every irresolute loop in P based at p is nullhomotopic in T, the loop ${\beta }_1\ast {\beta }_2^{-1}$ is irresolutely homotopic to the trivial irresolute path. Consequently, ${[\theta \ast {\beta }_1]}_G={[\theta \ast {\beta }_2]}_G$, proving injectivity.

Since $c_i{|}_{(P,\tilde{p})}$ is both injective and surjective, it is a semi-homeomorphism, showing that $(P,\tilde{p})$ is evenly irresolute-covered.

Finally, we show that $c_{i_{*}}{\pi }_1({\tilde{T}}_G,{\tilde{p}}_0)=G$. Let $\left[\gamma \right]\in {\pi }_1^{\mathrm{ir}}(T,p_0)$. Since $({\tilde{T}}_G,c_i)$ is an irresolute-covering space, there is a unique lift $\tilde{\gamma }$ of $\gamma$ starting at ${\tilde{p}}_0$. By construction, $\tilde{\gamma }\left(p\right)={\left[{\gamma }_p\right]}_G$, where ${\gamma }_p$ is the restriction of $\gamma$ to $[0,p]$. The path $\tilde{\gamma }$ is a loop in ${\tilde{T}}_G$ if and only if $\tilde{\gamma }\left(1\right)={\tilde{p}}_0$, which holds if and only if ${\left[\gamma \right]}_G$ is in G. Thus, $c_{i_{*}}{\pi }_1^{\mathrm{ir}}({\tilde{T}}_G,{\tilde{p}}_0)=G$, completing the proof. □

According to Theorem 8, the following theorem establishes a general result for topological groups with operations.

Theorem 11. 

Let $(G,\ast ,\tau )$ be an irresolute-topological group and let Γ be a subgroup of ${\pi }_1^{\mathrm{ir}}(G,e)$. Suppose that the underlying space of G is semi-connected, locally irresolute path-connected, and semi-locally s-simply connected. Let $c_i:({\tilde{G}}_{\Gamma },\tilde{e})\to (G,e)$ be the irresolute-covering map corresponding to Γ as a subgroup of the additive group ${\pi }_1^{\mathrm{ir}}(G,e)$ by Theorem 10 Then the operations of G lift to ${\tilde{G}}_{\Gamma }$, i.e., ${\tilde{G}}_{\Gamma }$ is an irresolute-topological group and $c_i:{\tilde{G}}_{\Gamma }\to G$ is a morphism of irresolute topological groups.

Proof. 

Let $P(G,e)$ be the set of all paths in G with initial point e. By the construction of ${\tilde{G}}_{\Gamma }$ in Section 2, ${\tilde{G}}_{\Gamma }$ consists of equivalence classes defined via $\Gamma$. The induced binary operations on $P(G,e)$ given by

$${\tilde{G}}_{\Gamma }\times {\tilde{G}}_{\Gamma }\to {\tilde{G}}_{\Gamma },{\left[\lambda \right]}_{\Gamma }\ast {\left[\zeta \right]}_{\Gamma }\mapsto {\left[\lambda \right]}_{\Gamma }\ast {\left[\zeta \right]}_{\Gamma }={[\lambda \ast \zeta ]}_{\Gamma }$$

and the unary operations given by

$$i:{\tilde{G}}_{\Gamma }\to {\tilde{G}}_{\Gamma },\left[\lambda \right]\mapsto {\left[\lambda \right]}^{-1}=\left[{\lambda }^{-1}\right]$$

are well-defined in ${\tilde{G}}_{\Gamma }$. Indeed, for the irresolute paths $\lambda ,\zeta ,{\lambda }_1,{\zeta }_1\in P(G,0)$ with $\lambda \left(1\right)={\lambda }_1\left(1\right)$ and $\zeta \left(1\right)={\zeta }_1\left(1\right)$, we have that

$$\langle (\lambda ★\zeta )\circ \overline{({\lambda }_1★{\zeta }_1)}\rangle$$

$$=\langle (\lambda ★\zeta )\circ (\overline{{\lambda }_1}★\overline{{\zeta }_1})\rangle \mathrm{by}\left(4\right)$$

$$=\langle (\lambda \circ \overline{{\lambda }_1})★(\zeta \circ \overline{{\zeta }_1})\rangle \mathrm{by}\left(3\right)$$

$$=\langle \lambda \circ \overline{{\lambda }_1}\rangle ★\langle \zeta \circ \overline{{\zeta }_1}\rangle .\mathrm{by}\left(1\right)$$

Thus, if ${\lambda }_1\in {\langle \lambda \rangle }_G$ and ${\zeta }_1\in {\langle \zeta \rangle }_G$, then

$$\langle \lambda \circ \overline{{\lambda }_1}\rangle \in G\mathrm{and}\langle \zeta \circ \overline{{\zeta }_1}\rangle \in G.$$

Since G is a subgroup of ${\pi }_1^{\mathrm{ir}}(G,0)$, we have

$$\langle \lambda \circ \overline{{\lambda }_1}\rangle ★\langle \zeta \circ \overline{{\zeta }_1}\rangle \in G.$$

Therefore, the binary operation (9) is well defined.

Similarly, for the paths $\lambda ,{\lambda }_1\in P(G,0)$ with $\lambda \left(1\right)={\lambda }_1\left(1\right)$, we have

$$\langle \left({\lambda }^{-1}\right)\circ \overline{{\left({\lambda }_1\right)}^{-1}}\rangle$$

$$=\langle \left({\lambda }^{-1}\right)\circ {\left(\overline{{\lambda }_1}\right)}^{-1})\rangle \mathrm{by}(5)$$

$$=\langle \left({(\lambda \circ \overline{{\lambda }_1})}^{-1}\right)\rangle \mathrm{by}\left(6\right)$$

$$=\langle \lambda \circ \overline{{\lambda }_1}){\rangle }^{-1}.\mathrm{by}\left(2\right)$$

Since G is a subgroup of ${\pi }_1^{\mathrm{ir}}(G,0)$, if $[\lambda \circ \overline{{\lambda }_1}]\in G$, then

$${\langle \lambda \circ \overline{{\lambda }_1}\rangle }^{-1}\in G.$$

Hence, the unary operation (10) is also well defined.

Since $\Gamma$ is a subgroup of ${\pi }_1^{\mathrm{ir}}(G,e)$, these operations satisfy the conditions of an irresolute topological group. We need to prove that ${\tilde{G}}_{\Gamma }$ is an irresolute topological group, and $c_i$ is a morphism of irresolute topological groups. Let ${\left[\lambda \right]}_G,{\left[\zeta \right]}_G^{-1}\in {\tilde{G}}_{\Gamma }$ and $(R,{[\lambda ★{\zeta }^{-1}]}_G)$ be a basic open neighborhood of ${[\lambda ★{\zeta }^{-1}]}_G$. Here, R is an open neighborhood of $(\lambda ★{\zeta }^{-1})\left(1\right)=\lambda \left(1\right)★{\zeta }^{-1}\left(1\right)$. Since the operations $★:G\times G\to G$ are continuous, there are open neighborhoods P and O of $\lambda \left(1\right)$ and $\zeta \left(1\right)$, respectively, in G such that $P★O^{-1}\subseteq R$. Therefore, $(P,{\left[\lambda \right]}_G)$ and $(O,{\left[\zeta \right]}_G)$ are respectively base open neighborhoods of ${\left[\lambda \right]}_G$ and ${\left[\zeta \right]}_G$, and

$$(P,{\left[\lambda \right]}_G)★{(O,{\left[\zeta \right]}_G)}^{-1}\subseteq (R,{[\lambda ★\zeta ]}_G).$$

Thus, the necessary conditions for an irresolute-topological group are satisfied. The map $c_i:{\tilde{G}}_{\Gamma }\to G$ defined by $c_i\left({\left[\lambda \right]}_{\Gamma }\right)=\lambda \left(1\right)$ preserves the operations and is a morphism of irresolute topological groups. □

Theorem 12. 

Suppose that T is an irresolute-topological space, where the underlying space is semi-connected, locally irresolute path-connected, and semi-locally s-simply connected. Let

$$c_i:(\tilde{T},\tilde{e})\to (T,e)$$

be an irresolute-covering map such that $\tilde{T}$ is irresolute path-connected, and let G be the characteristic group associated with $c_i$, which is a subgroup of ${\pi }_1^{\mathrm{ir}}(T,e)$. Then, the group operations on T lift naturally to $\tilde{T}$.

Proof. 

By assumption, the characteristic group G of the irresolute-covering map

$$c_i:(\tilde{T},\tilde{e})\to (T,e)$$

is a subgroup of the fundamental group ${\pi }_1^{\mathrm{ir}}(T,e)$. Consequently, following the result stated in Remark 2, we can consider $\tilde{T}$ as ${\tilde{T}}_G$. From Theorem 11, it follows that the group operations defined on T extend to $\tilde{T}$ as needed. □

In particular, in Theorem 11, if the subgroup G of ${\pi }_1^{\mathrm{ir}}(T,e)$ is chosen to be the singleton, then the following corollary is obtained.

Corollary 3. 

Under the same conditions as Theorem 12, if the characteristic subgroup G of ${\pi }_1^{\mathrm{ir}}(T,e)$ is trivial (i.e., a singleton set), then the irresolute-covering map

$$c_i:(\tilde{T},\tilde{e})\to (T,e)$$

becomes a universal irresolute-covering map. In this case, the group operations on T lift completely to $\tilde{T}$.

5. Conclusions

In this study, we introduced the concept of an irresolute path and defined irresolute path connectedness. We then explored irresolute path connected components and their properties. Our main objective was to reconstruct the fundamental group using irresolute functions and to present a different approach to homotopy. This new perspective provides a solid foundation for further studies in this field.

Covering spaces are excellent tools for understanding the fundamental group (and even higher homotopy groups), especially universal covering spaces. For this reason, we redefined the concept of covering spaces using irresolute functions. We obtained properties similar to those in topological spaces; however, it is important to note that the concepts of covering and irresolute covering are independent of each other. Furthermore, using these properties, we demonstrated that the structure of an irresolute topological group lifts to its irresolute covering space.

In future studies, we will use these definitions to reconstruct the category of covering spaces and extend it to irresolute groups. Thus, we aim to establish and prove certain categorical equivalences.