# Topology on Soft Continuous Function Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

- (1)
- $\Phi ,\stackrel{\sim}{X}$ belongs to $\tau ;$
- (2)
- the union of any number of soft sets in τ belongs to $\tau ;$
- (3)
- the intersection of any two soft sets in τ belongs to $\tau .$

**Definition**

**11.**

**Proposition**

**1.**

**Definition**

**12.**

**Definition**

**13.**

**Definition**

**14.**

**Definition**

**15.**

**Definition**

**16.**

**Definition**

**17.**

**Definition**

**18.**

**Definition**

**19.**

**Definition 20.**${\left\{({\phi}_{i},{\psi}_{i}):(X,\tau ,E)\to \left({Y}_{i},{\tau}_{i},{E}_{i}\right)\right\}}_{i\in \Delta}$ is a family of soft mappings, and ${\left\{\left({Y}_{i},{\tau}_{i},{E}_{i}\right)\right\}}_{i\in \Delta}$ is a family of soft topological spaces [14]. Then, the topology τ generated from the subbase $\delta \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left\{{({\phi}_{i},{\psi}_{i})}_{i\in \Delta}^{-1}({F}_{i},{E}_{i}):({F}_{i},{E}_{i})\in {\tau}_{i},i\in \Delta \right\}$ is called the soft topology (or initial soft topology) induced by the family of soft mappings ${\left\{({\phi}_{i},{\psi}_{i})\right\}}_{i\in \Delta}.$

**Definition**

**21.**

## 3. Topology on Soft Continuous Function Spaces

**Theorem**

**2.**

**Proof.**

**Definition**

**22.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Definition**

**23.**

**Example**

**5.**

**Remark**

**5.**

**Proposition**

**6.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Molodtsov, D. Soft Set Theory-First Results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] - Maji, P.K.; Bismas, R.; Roy, A.R. Soft Set Theory. Comput. Math. Appl.
**2003**, 45, 555–562. [Google Scholar] [CrossRef] - Maji, P.K.; Roy, A.R. An Application of Soft Sets in a Decision Making Problem. Comput. Math. Appl.
**2002**, 44, 1077–1083. [Google Scholar] [CrossRef] - Aktas, H.; Çağman, N. Soft Sets and Soft Group. Inf. Sci.
**2007**, 177, 2726–2735. [Google Scholar] [CrossRef] - Acar, U.; Koyuncu, F.; Tanay, B. Soft Sets and Soft Rings. Comput. Math. Appl.
**2010**, 59, 3458–3463. [Google Scholar] [CrossRef] - Feng, F.; Jun, Y.B.; Zhao, X. Soft Semirings. Comput. Math. Appl.
**2008**, 56, 2621–2628. [Google Scholar] [CrossRef] - Shabir, M.; İrfan Ali, M. Soft ideals and generalized fuzzy ideals in semigroups. New Math. Nat. Comput.
**2009**, 5, 599–615. [Google Scholar] [CrossRef] - Sun, Q.M.; Zhang, L.; Liu, J. Soft Sets and Soft Modules. Lect. Notes Comput. Sci.
**2008**, 5009, 403–409. [Google Scholar] - Gunduz, Ç.; Bayramov, S. Fuzzy Soft modules. Int. Mat. Forum
**2011**, 6, 517–527. [Google Scholar] - Gunduz, Ç.; Bayramov, S. Intuitionistic Fuzzy Soft Modules. Comput. Math. Appl.
**2011**, 62, 2480–2486. [Google Scholar] [CrossRef] - Ozturk, T.Y.; Bayramov, S. Category of chain complexes of soft modules. Int. Math. Forum
**2012**, 7, 981–992. [Google Scholar] - Ozturk, T.Y.; Gunduz, C.; Bayramov, S. Inverse and direct systems of soft modules. Ann. Fuzzy Math. Inf.
**2013**, 5, 73–85. [Google Scholar] - Shabir, M.; Naz, M. On soft topological spaces. Comput. Math. Appl.
**2011**, 61, 1786–1799. [Google Scholar] [CrossRef] - Aygunoğlu, A.; Agun, A. Some notes on soft topological spaces. Neural Comput. Appl.
**2012**, 21, 113–119. [Google Scholar] [CrossRef] - Çağman, N.; Karataş, S.; Enginoğlu, S. Soft topology. Comput. Math. Appl.
**2011**, 351–358. [Google Scholar] [CrossRef] - Min, W.K. A note on soft topological spaces. Comput. Math. Appl.
**2011**, 62, 3524–3528. [Google Scholar] [CrossRef] - Shabir, H.; Bashir, A. Some properties of soft topological spaces. Comput. Math. Appl.
**2011**, 62, 4058–4067. [Google Scholar] - Zorlutuna, İ.; Akdağ, M.; Min, W.K.; Atmaca, S. Remarks on soft topological spaces. Ann. Fuzzy Math. Inf.
**2012**, 3, 171–185. [Google Scholar] - Bayramov, S.; Gunduz, C. Soft locally compact spaces and soft paracompact spaces. J. Math. Syst. Sci.
**2013**, 3, 122–130. [Google Scholar] - Gunduz, A.C.; Sonmez, A.; Çakallı, H. On soft Mappings. arXiv, 2013; arXiv:1305.4545. [Google Scholar]
- Ozturk, T.Y.; Bayramov, S. Soft mappings space. Sci. World J.
**2014**, 2014. [Google Scholar] [CrossRef] [PubMed]

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Öztürk, T.Y.; Bayramov, S.
Topology on Soft Continuous Function Spaces. *Math. Comput. Appl.* **2017**, *22*, 32.
https://doi.org/10.3390/mca22020032

**AMA Style**

Öztürk TY, Bayramov S.
Topology on Soft Continuous Function Spaces. *Mathematical and Computational Applications*. 2017; 22(2):32.
https://doi.org/10.3390/mca22020032

**Chicago/Turabian Style**

Öztürk, Taha Yasin, and Sadi Bayramov.
2017. "Topology on Soft Continuous Function Spaces" *Mathematical and Computational Applications* 22, no. 2: 32.
https://doi.org/10.3390/mca22020032