Fuzzy Semi-Metric Spaces

The T1-spaces induced by the fuzzy semi-metric spaces endowed with the special kind of triangle inequality are investigated in this paper. The limits in fuzzy semi-metric spaces are also studied to demonstrate the consistency of limit concepts in the induced topologies.


Introduction
Given a universal set X, for any x, y ∈ X, let d(x, y) be a fuzzy subset of R + with membership function ξ d(x,y) : R + → [0, 1], where the value ξ d(x,y) (t) means that the membership degree of the distance between x and y is equal to t. Kaleva and Seikkala [1] proposed the fuzzy metric space by defining a function M * : X × X × [0, ∞) → [0, 1] as follows: M * (x, y, t) = ξ d(x,y) (t). ( On the other hand, inspired by the Menger space that is a special kind of probabilistic metric space (by referring to Schweizer and Sklar [2][3][4], Hadžić and Pap [5] and Chang et al. [6]), Kramosil and Michalek [7] proposed another concept of fuzzy metric space.
Let X be a nonempty universal set, let * be a t-norm, and let M be a mapping defined on X × X × [0, ∞) into [0, 1].The 3-tuple (X, M, * ) is called a fuzzy metric space if and only if the following conditions are satisfied:

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for any x, y ∈ X, M(x, y, t) = 1 for all t > 0 if and only if x = y; • M(x, y, 0) = 0 for all x, y ∈ X; • M(x, y, t) = M(y, x, t) for all x, y ∈ X and t ≥ 0; • M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t ≥ 0 (the so-called triangle inequality).
The mapping M in fuzzy metric space (X, M, * ) can be regarded as a membership function of a fuzzy subset of X × X × [0, ∞).Sometimes, M is called a fuzzy metric of the space (X, M, * ).According to the first and second conditions of fuzzy metric space, the mapping M(x, y, t) can be interpreted as the membership degree of the distance that is less than t between x and y.Therefore, the meanings of M and M * defined in Equation (1) are different.
George and Veeramani [8,9] studied some properties of fuzzy metric spaces.Gregori and Romaguera [10][11][12] also extended their research to study the properties of fuzzy metric spaces and fuzzy quasi-metric spaces.In particular, Gregori and Romaguera [11] proposed the fuzzy quasi-metric spaces in which the symmetric condition was not assumed.In this paper, we study the so-called fuzzy semi-metric space without assuming the symmetric condition.The main difference is that four forms of triangle inequalities that were not addressed in Gregori and Romaguera [11] are considered in this paper.Another difference is that the t-norm in Gregori and Romaguera [11] was assumed to be continuous.However, the assumption of continuity for t-norm is relaxed in this paper.
The Hausdorff topology induced by the fuzzy metric space was studied in Wu [13], and the concept of fuzzy semi-metric space was considered in Wu [14].In this paper, we shall extend to study the T 1 -spaces induced by the fuzzy semi-metric spaces that is endowed with special kind of triangle inequality.Roughly speaking, the fuzzy semi-metric space does not assume the symmetric condition M(x, y, t) = M(y, x, t).In this case, there are four kinds of triangle inequalities that can be considered, which will be presented in Definition 2. We shall induce the T 1 -spaces from the fuzzy semi-metric space based on a special kind of triangle inequality, which will generalize the results obtained in Wu [13].On the other hand, since the symmetric condition is not satisfied in the fuzzy semi-metric space, three kinds of limit concepts will also be considered in this paper.Furthermore, we shall prove the consistency of limit concepts in the induced topologies.
This paper is organized as follows.In Section 2, the basic properties of t-norm are presented that will be used for the further discussion.In Section 3, we propose the fuzzy semi-metric space that is endowed with four kinds of triangle inequalities.In Section 4, we induce the T 1 -space from a given fuzzy semi-metric space endowed with a special kind of triangle inequality.In Section 5, three kinds of limits in fuzzy semi-metric space will be considered.We also present the consistency of limit concepts in the induced topologies.

Properties of t-Norm
We first recall the concept of triangular norm (i.e., t-norm).We consider the function * : . The function * is called a t-norm if and only if the following conditions are satisfied: From the third condition, it follows that, for any a ∈ [0, 1], we have 0 * a ≤ 0 * 1.From the first condition, we also have 0 * 1 = 0, which implies 0 * a = 0.The following proposition from Wu [13] will be useful for further study Proposition 1.By the commutativity of t-norm, if the t-norm is continuous with respect to the first component (resp.second component), then it is also continuous with respect to the second component (resp.first component).In other words, for any fixed a ∈ [0, 1], if the function f (x) = a * x (resp.f (x) = x * a) is continuous, then the function g(x) = x * a (resp.g(x) = a * x) is continuous.Similarly, if the t-norm is left-continuous (resp.right-continuous) with respect to the first or second component, then it is also left-continuous (resp.right-continuous) with respect to each component.
We first provide some properties that will be used in the subsequent discussion.
Proposition 2. We have the following properties: Proof.To prove part (i), since a n → a as n → ∞, there exist an increasing sequence {p n } ∞ n=1 and a decreasing sequence {q n } ∞ n=1 such that p n ↑ a and q n ↓ a satisfying p n ≤ a n ≤ q n .In addition, there exists an increasing sequence {r n } ∞ n=1 and a decreasing sequence {s n } ∞ n=1 such that r n ↑ b and s n ↓ b satisfying r n ≤ b n ≤ s n .By Remark 1, we see that the t-norm is continuous with respect to each component.Given any > 0, using the continuity of t-norm at b with respect to the second component, there exists In addition, using the continuity of t-norm at a with respect to the first component, there exists According to Equation (2) and using the increasing property of t-norm, for n ≥ n 0 , we have In addition, according to Equation (3), for m ≥ n 1 and n ≥ n 0 , we have By taking n 2 = max{n 0 , n 1 }, from Equations ( 4) and ( 6), we obtain that n ≥ n To prove part (ii), we note that there exist two increasing sequences {p n } ∞ n=1 and {r n } ∞ n=1 such that p n ↑ a and r n ↑ b satisfying p n ≤ a n and r n ≤ b n .By Remark 1, we see that the t-norm is left continuous with respect to each component.Given any > 0, using the left-continuity of t-norm at b with respect to the second component, there exists In addition, using the left-continuity of t-norm at a with respect to the first component, there exists Using the increasing property of t-norm, for m ≥ n 1 and n ≥ n 0 , we have This shows the desired convergence.Part (iii) can be similarly proved, and the proof is complete.
The associativity of t-norm says that the operation a 1 * a 2 * • • • * a p is well-defined for p ≥ 2. The following proposition from Wu [13] will be useful for further study.Proposition 3. Suppose that the t-norm * is left-continuous at 1 with respect to the first or second component.We have the following properties: (i) For any a, b ∈ (0, 1) with a > b, there exists r ∈ (0, 1) such that a * r ≥ b.

Fuzzy Semi-Metric Space
In the sequel, we shall define the concept of fuzzy semi-metric space without considering the symmetric condition.Because of lacking symmetry, the concept of triangle inequality should be carefully interpreted.Therefore, we propose four kinds of triangle inequalities.Definition 1.Let X be a nonempty universal set, and let M be a mapping defined on X × X × [0, ∞) into [0, 1].Then, (X, M) is called a fuzzy semi-metric space if and only if the following conditions are satisfied: We say that M satisfies the symmetric condition if and only if M(x, y, t) = M(y, x, t) for all x, y ∈ X and t > 0. We say that M satisfies the strongly symmetric condition if and only if M(x, y, t) = M(y, x, t) for all x, y ∈ X and t ≥ 0.
We remark that the first condition says that M(x, x, t) = 1 for all t > 0. However, the value of M(x, x, 0) is free.Recall that the mapping M(x, y, t) is interpreted as the membership degree of the distance that is less than t between x and y.Therefore, M(x, x, t) = 1 for all t > 0 means that the distance that is less than t > 0 between x and x is always true.The second condition says that M(x, y, 0) = 0 for x = y, which can be similarly realized that the distance that is less than 0 between two distinct elements x and y is impossible.Definition 2. Let X be a nonempty universal set, let * be a t-norm, and let M be a mapping defined on

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied: M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied: M(x, y, t) * M(z, y, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied: M(y, x, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.

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We say that M satisfies the -triangle inequality if and only if the following inequality is satisfied: M(y, x, t) * M(z, y, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0.
We say that M satisfies the strong •-triangle inequality for • ∈ { , , , } when s, t > 0 is replaced by s, t ≥ 0. Remark 1.It is obvious that if the mapping M satisfies the symmetric condition, then the concepts of -triangle inequality, -triangle inequality, -triangle inequality and -triangle inequality are all equivalent.
Example 1.Let X be a universal set, and let d : X × X → R + satisfy the following conditions: Note that we do not assume d(x, y) = d(y, x).For example, let X = [0, 1].We define Then, d(x, y) = d(y, x) and the above three conditions are satisfied.Now, we take t-norm * as a * b = ab and define It is clear to see that M(x, y, t) = M(y, x, t) for t > 0, since d(x, y) = d(y, x).We are going to claim that (X, M, * ) is a fuzzy semi-metric space satisfying the -triangle inequality.For t > 0 and M(x, y, t) = 1, we have t = t + d(x, y), which says that d(x, y) = 0, i.e., x = y.Next, we are going to check the -triangle inequality.For s > 0 and t > 0, we first have Then, we obtain This shows that (X, M, * ) defined above is indeed a fuzzy semi-metric space satisfying the -triangle inequality.
Given a fuzzy semi-metric space (X, M), when we say that the mapping M satisfies some kinds of (strong) triangle inequalities, it implicitly means that the t-norm is considered in (X, M).
Definition 3. Let (X, M) be a fuzzy semi-metric space.

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We say that M is nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(x, y, t 2 ) for t 1 > t 2 > 0. We say that M is strongly nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(x, y, t 2 ) for t 1 > t 2 ≥ 0.

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We say that M is symmetrically nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(y, x, t 2 ) for t 1 > t 2 > 0. We say that M is symmetrically strongly nondecreasing if and only if, given any fixed x, y ∈ X, M(x, y, t 1 ) ≥ M(y, x, t 2 ) for t 1 > t 2 ≥ 0.
Proposition 4. Let (X, M) be a fuzzy semi-metric space.Then, we have the following properties: (i) If M satisfies the -triangle inequality, then M is nondecreasing.
(ii) If M satisfies the -triangle inequality or the -triangle inequality, then M is both nondecreasing and symmetrically nondecreasing.(iii) If M satisfies the -triangle inequality, then M is symmetrically nondecreasing.
Proof.Given any fixed x, y ∈ X, for t 1 > t 2 > 0, we have the following inequalities.
• Suppose that M satisfies the -triangle inequality.Then, • Suppose that M satisfies the -triangle inequality.Then, This completes the proof.
Definition 4. Let (X, M) be a fuzzy semi-metric space.

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We say that M is left-continuous with respect to the distance at t 0 > 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t 0 − t < δ implies |M(x, y, t) − M(x, y, t 0 )| < ; that is, the mapping M(x, y, •) : (0, ∞) → [0, 1] is left-continuous at t 0 .We say that M is left-continuous with respect to the distance on (0, ∞) if and only if the mapping M(x, y, •) is left-continuous on (0, ∞) for any fixed x, y ∈ X.

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We say that M is right-continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t − t 0 < δ implies |M(x, y, t) − M(x, y, t 0 )| < ; that is, the mapping M(x, y, •) : (0, ∞) → [0, 1] is right-continuous at t 0 .We say that M is right-continuous with respect to the distance on [0, ∞) if and only if the mapping M(x, y, •) is left-continuous on [0, ∞) for any fixed x, y ∈ X.

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We say that M is continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that |t − t 0 | < δ implies |M(x, y, t) − M(x, y, t 0 )| < ; that is, the mapping M(x, y, •) : (0, ∞) → [0, 1] is continuous at t 0 .We say that M is continuous with respect to the distance on [0, ∞) if and only if the mapping M(x, y, •) is continuous on [0, ∞) for any fixed x, y ∈ X.

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We say that M is symmetrically left-continuous with respect to the distance at t 0 > 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t 0 − t < δ implies |M(x, y, t) − M(y, x, t 0 )| < .We say that M is symmetrically left-continuous with respect to the distance on (0, ∞) if and only if it is symmetrically left-continuous with respect to the distance at each t > 0.

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We say that M is symmetrically right-continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists δ > 0 such that 0 < t − t 0 < δ implies |M(x, y, t) − M(y, x, t 0 )| < .We say that M is symmetrically right-continuous with respect to the distance on [0, ∞) if and only if it is symmetrically right-continuous with respect to the distance at each t ≥ 0.

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We say that M is symmetrically continuous with respect to the distance at t 0 ≥ 0 if and only if, for any fixed x, y ∈ X, given any > 0, there exists We say that M is symmetrically continuous with respect to the distance on [0, ∞) if and only if it is symmetrically continuous with respect to the distance at each t ≥ 0.
Proposition 5. Let (X, M) be a fuzzy semi-metric space such that the •-triangle inequality is satisfied for • ∈ { , , }.Then, we have the following properties: (i) Suppose that M is left-continuous or symmetrically left-continuous with respect to the distance at t > 0.
Then M(x, y, t) = M(y, x, t).In other words, if M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞).Then M satisfies the symmetric condition.(ii) Suppose that M is right-continuous or symmetrically right-continuous with respect to the distance at t ≥ 0.
Then M(x, y, t) = M(y, x, t).In other words, if M is right-continuous or symmetrically right-continuous with respect to the distance on [0, ∞).Then M satisfies the strongly symmetric condition.
Proof.To prove part (i), given any t > 0, there exists n t ∈ N satisfying t − 1 n t > 0. We consider the following cases:

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Suppose that the -triangle inequality is satisfied.Then, Using the left-continuity of M, it follows that M(y, x, t) ≤ M(x, y, t) and M(x, y, t) ≤ M(y, x, t) by taking n t → ∞.This shows that M(x, y, t) = M(y, x, t) for all t > 0. On the other hand, we also have Using the symmetric left-continuity of M, it follows that M(y, x, t) ≤ M(x, y, t) and M(x, y, t) ≤ M(y, x, t) by taking n t → ∞.This shows that M(x, y, t) = M(y, x, t) for all t > 0.

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Suppose that the -triangle inequality is satisfied.Then, and The left-continuity of M shows that M(x, y, t) = M(y, x, t) for all t > 0. We can similarly obtain the desired result using the symmetric left-continuity of M.

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Suppose that the -triangle inequality is satisfied.Then, this is the same situation as the -triangle inequality.
To prove part (ii), given any t ≥ 0 and n ∈ N, we consider the following cases.
• Suppose that the -triangle inequality is satisfied.Then, The right-continuity of M shows that M(x, y, t) = M(y, x, t) for all t ≥ 0. We can similarly obtain the desired result using the symmetric right-continuity of M.

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Suppose that the -triangle inequality is satisfied.Then, The right-continuity of M shows that M(x, y, t) = M(y, x, t) for all t ≥ 0. We can similarly obtain the desired result using the symmetric right-continuity of M.

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Suppose that the -triangle inequality is satisfied.Then, this is the same situation as the -triangle inequality.
This completes the proof.
From Proposition 5, if M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞), or right-continuous and or symmetrically right-continuous with respect to the distance on on (0, ∞], then we can just consider the -triangle inequality.Definition 5. Let (X, M) be a fuzzy semi-metric space.Given t > 0 and 0 < r < 1, the (r, t)-balls centered at x are denoted and defined by B (x, r, t) = {y ∈ X : M(x, y, t) > 1 − r} and B (x, r, t) = {y ∈ X : M(y, x, t) > 1 − r} .
Let B denote the family of all (r, t)-balls B (x, r, t), and let B denote the family of all (r, t)-balls B (x, r, t).
It is clearly that if the symmetric condition for M is satisfied, then In this case, we simply write B(x, r, t) to denote the (r, t)-balls centered at x, and write B to denote the family of all (r, t)-balls B(x, r, t).
We also see that B (x, r, t) = ∅ and B (x, r, t) = ∅, since x ∈ B (x, r, t) and x ∈ B (x, r, t) by the fact of M(x, x, t) = 1 for all t > 0. Since 0 < r < 1, it is obvious that if M(x, y, t) = 0, then y ∈ B (x, r, t).In other words, if y ∈ B (x, r, t), then M(x, y, t) > 0. Similarly, if y ∈ B (x, r, t), then M(y, x, t) > 0. Proposition 8. Let (X, M) be a fuzzy semi-metric space.
(i) For each x ∈ X, we have x ∈ B (x, r, t) ∈ B and x ∈ B (x, r, t) ∈ B .(ii) If x = y, then there exist B (x, r, t) and B (x, r, t) such that y ∈ B (x, r, t) and y ∈ B (x, r, t).
(i) Suppose that M satisfies the •-triangle for • ∈ { , , }.Then, the following statements hold true: (ii) Suppose that M satisfies the •-triangle for • ∈ { , , }.Then, the following statements hold true: Proof.To prove part (i), it suffices to prove the first case.We take n ∈ N such that 1/n ≤ min{r, t}.
Then, for y ∈ B (x, 1/n, 1/n), using parts (i) and (ii) of Proposition 4, we have which says that y ∈ B (x, r, t).Part (ii) can be similarly obtained by using parts (ii) and (iii) of Proposition 4, and the following inequalities: This completes the proof.
Proposition 10. (Left-Continuity for M) Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that the following conditions are satisfied: • M is left-continuous with respect to the distance on (0, ∞); • the t-norm * is left-continuous at 1 with respect to the first or second component.
Suppose that M satisfies the -triangle inequality.Then, we have the following inclusions: (i) Given any y ∈ B (x, r, t), there exists B (y, r, t) such that B (y, r, t) ⊆ B (x, r, t).
(ii) Given any y ∈ B (x, r, t), there exists B (y, r, t) such that B (y, r, t) ⊆ B (x, r, t).
To prove part (i), for y ∈ B (x, r, t) and z ∈ B (y, r, t), we have By the -triangle inequality, we also have This shows that z ∈ B (x, r, t).Therefore, we obtain the inclusion B (y, r, t) ⊆ B (x, r, t).
To prove part (ii), for y ∈ B (x, r, t) and z ∈ B (y, r, t), we have By the -triangle inequality, we also have This shows that z ∈ B (x, r, t).Therefore, we obtain the inclusion B (y, r, t) ⊆ B (x, r, t).This completes the proof.
According to Proposition 5, since M is assumed to be left-continuous with respect to the distance on (0, ∞), it is not necessarily to consider the •-triangle inequality for • ∈ { , , } in Proposition 10.Proposition 11. (Symmetric Left-Continuity for M) Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that the following conditions are satisfied:

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M is symmetrically left-continuous with respect to the distance on (0, ∞); • the t-norm * is left-continuous at 1 with respect to the first or second component.
Suppose that M satisfies the -triangle inequality.Then, we have the following inclusions: (i) Given any y ∈ B (x, r, t), there exists B (y, r, t) such that B (y, r, t) ⊆ B (x, r, t).
(ii) Given any y ∈ B (x, r, t), there exists B (y, r, t) such that B (y, r, t) ⊆ B (x, r, t).
To prove part (i), for y ∈ B (x, r, t) and z ∈ B (y, r, t), we have By the -triangle inequality, we have This shows that z ∈ B (x, r, t).Therefore, we obtain the inclusion B (y, r, t) ⊆ B (x, r, t).
To prove part (ii), for y ∈ B (x, r, t) and z ∈ B (y, r, t), we have By the -triangle inequality, we have This shows that z ∈ B (x, r, t).Therefore, we obtain the inclusion B (y, r, t) ⊆ B (x, r, t).This completes the proof.
According to Proposition 5, since M is assumed to be symmetrically left-continuous with respect to the distance on (0, ∞), it is not necessarily to consider the •-triangle inequality for • ∈ { , , } in Proposition 11.
Proposition 12. (Left-Continuity for M) Let (X, M) be a fuzzy semi-metric space along with a t-norm * such that the following conditions are satisfied:

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M is left-continuous with respect to the distance on (0, ∞); • the t-norm * is left-continuous at 1 with respect to the first or second component.
We take t 3 = min{ t1 , t2 } and r 3 = min{r 1 , r2 }.Then, for y ∈ B (x, r 3 , t 3 ), using part (i) of Proposition 4, we have Since M satisfies the -triangle inequality, it follows that which is a contradiction.On the other hand, for Since M satisfies the -triangle inequality, it follows that which is a contradiction.
This completes the proof.
Theorem 1.Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous at 1 with respect to the first or second component.Suppose that M is left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞), and that M satisfies the -triangle inequality.
(i) We define τ = {O ⊆ X : x ∈ O if and only if there exist t > 0 and r ∈ (0, 1) such that B (x, r, t) ⊆ O } .
Then, the family B induces a T 1 -space (X, τ ) such that B is a base for the topology τ , in which O ∈ τ if and only if, for each x ∈ O , there exist t > 0 and r ∈ (0, 1) such that B (x, r, t) ⊆ O .(ii) We define τ = {O ⊆ X : x ∈ O if and only if there exist t > 0 and r ∈ (0, 1) such that B (x, r, t) ⊆ O } .
Then, the family B induces a T 1 -space (X, τ ) such that B is a base for the topology τ , in which O ∈ τ if and only if, for each x ∈ O , there exist t > 0 and r ∈ (0, 1) such that B (x, r, t) ⊆ O .
Proof.Using part (i) of Proposition 8, part (i) of Proposition 12 and part (i) of Proposition 13, we see that τ is a topology such that B is a base for τ .Part (ii) of Proposition 8 says that (X, τ ) is a T 1 -space.Part (i) of Proposition 9 says that there exist countable local bases at each x ∈ X for τ and τ , respectively, which also says that τ and τ satisfy the first axiom of countability.We can similarly obtain the desired results regarding the topology τ .This completes the proof.
According to Proposition 5, since M is assumed to be left-continuous or symmetrically left-continuous with respect to the distance on (0, ∞), it follows that the topologies obtained in Wu [13] are still valid when we consider the •-triangle inequality for • ∈ { , , }.
Proposition 15.Let (X, M) be a fuzzy semi-metric space along with a t-norm * that is left-continuous at 1 with respect to the first or second component.Suppose that M is left-continuous with respect to the distance on (0, ∞), and that M satisfies the -triangle inequality.Then, regarding the T 1 -spaces (X, τ ) and (X, τ ), B (x, r, t) is a τ -open set and B (x, r, t) is a τ -open set.

Conclusions
In fuzzy metric space, the triangle inequality plays an important role.In general, since the symmetric condition is not necessarily to be satisfied, the so-called fuzzy semi-metric space is proposed in this paper.In this situation, four different types of triangle inequalities are proposed and studied.The main purpose of this paper is to establish the T 1 -spaces that are induced by the fuzzy semi-metric spaces along with the special kind of triangle inequality.
On the other hand, the limit concepts in fuzzy semi-metric space are also proposed and studied in this paper.Since the symmetric condition is not satisfied, three kinds of limits in fuzzy semi-metric space are considered.The concepts of uniqueness for the limits are also studied.Finally, we present the consistency of limit concepts in the induced T 1 -spaces.

( i )
Given any fixed a, b ∈ [0, 1], suppose that the t-norm * is continuous at a and b with respect to the first or second component.If {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] such that a n → a and b n → b as n → ∞, then a n * b n → a * b as n → ∞. (ii) Given any fixed a, b ∈ (0, 1], suppose that the t-norm * is left-continuous at a and b with respect to the first or second component.If {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] such that a n → a− and b n → b− as n → ∞, then a n * b n → a * b as n → ∞. (iii) Given any fixed a, b ∈ [0, 1), suppose that the t-norm * is right-continuous at a and b with respect to the first or second component.If {a n } ∞ n=1 and {b n } ∞ n=1 are two sequences in [0, 1] such that a n → a+ and b n → b+ as n → ∞, then a n * b n → a * b as n → ∞.