# On Slight Omega Continuity and Irresoluteness between Generalized Topological Spaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Slightly $\left(\mathbf{\mu},\mathbf{\sigma}\right)$-Continuous Functions

**Definition**

**1.**

**Theorem**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Theorem**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Definition**

**3.**

**Proposition**

**2.**

**Proof.**

**Example**

**6.**

**Theorem**

**3.**

**Proof.**

**Definition**

**4.**

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**Theorem**

**5.**

**Definition**

**5.**

**Theorem**

**6.**

**Remark**

**2.**

**Definition**

**6.**

**Theorem**

**7.**

**Proof.**

**Definition**

**7.**

**Definition**

**8.**

**Proposition**

**3.**

**Corollary**

**1.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Remark**

**3.**

**Theorem**

**10.**

**Proof.**

**Example**

**7.**

**Example**

**8.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

## 3. Slightly $\mathbf{\omega}$-$\left(\mathbf{\mu},\mathbf{\sigma}\right)$-Continuous Functions

**Definition**

**11.**

**Definition**

**12.**

**Example**

**9.**

**Proof.**

**Example**

**10.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Example**

**11.**

**Theorem**

**14.**

**Proof.**

**Example**

**12.**

**Theorem**

**15.**

**Proof.**

**Theorem**

**16.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Definition**

**13.**

**Theorem**

**18.**

**Proof.**

**Theorem**

**19.**

**Proof.**

**Theorem**

**20.**

**Definition**

**14.**

**Theorem**

**21.**

**Proof.**

**Example**

**13.**

**Proof.**

**Example**

**14.**

**Theorem**

**22.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**15.**

**Theorem**

**23.**

**Proof.**

## 4. Slightly $\mathbf{\omega}$-$\left(\mathbf{\mu},\mathbf{\sigma}\right)$-Irresolute Functions

**Definition**

**15.**

**Definition**

**16.**

**Example**

**16.**

**Proof.**

**Theorem**

**24.**

**Proof.**

**Example**

**17.**

**Example**

**18.**

**Theorem**

**25.**

**Proof.**

**Corollary**

**3.**

**Definition**

**17.**

**Theorem**

**26.**

## 5. Covering Properties

**Definition**

**18.**

**Remark**

**4.**

**Definition**

**19.**

**Definition**

**20.**

**Definition**

**21.**

**Definition**

**22.**

**Remark**

**5.**

**Definition**

**23.**

**Theorem**

**27.**

**Proof.**

**Claim.**$\mathcal{H}$ is a cover of B by ${\mu}_{2}$-clopen sets.

**Proof**

**of Claim.**

**Corollary**

**4.**

**Definition**

**24.**

**Definition**

**25.**

**Remark**

**6.**

**Theorem**

**28.**

**Proof.**

**Claim.**$\mathcal{H}$ is a countable cover of B by ${\mu}_{2}$-clopen sets.

**Proof**

**of Claim.**

**Corollary**

**5.**

**Theorem**

**29.**

**Proof.**

**Claim.**$\mathcal{H}$ is a cover of B by ${\mu}_{2}$-open sets.

**Proof**

**of Claim.**

**Corollary**

**6.**

**Corollary**

**7.**

**Corollary**

**8.**

**Definition**

**26.**

**Definition**

**27.**

**Remark**

**7.**

**Theorem**

**30.**

**Proof.**

**Claim.**$\mathcal{H}$ is a cover of B by ${\mu}_{2}$-clopen sets.

**Proof**

**of Claim.**

**Corollary**

**9.**

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Al Ghour, S.; Al-Nimer, A.
On Slight Omega Continuity and Irresoluteness between Generalized Topological Spaces. *Symmetry* **2020**, *12*, 780.
https://doi.org/10.3390/sym12050780

**AMA Style**

Al Ghour S, Al-Nimer A.
On Slight Omega Continuity and Irresoluteness between Generalized Topological Spaces. *Symmetry*. 2020; 12(5):780.
https://doi.org/10.3390/sym12050780

**Chicago/Turabian Style**

Al Ghour, Samer, and Abeer Al-Nimer.
2020. "On Slight Omega Continuity and Irresoluteness between Generalized Topological Spaces" *Symmetry* 12, no. 5: 780.
https://doi.org/10.3390/sym12050780