# Categories of Open Sets in Generalized Primal Topological Spaces

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. $\mathbf{GPT}$ Space

**Definition 1.**

**Remark 1.**

**Definition 2.**

**Remark 2.**

**Definition 3.**

**Theorem 1.**

**Theorem 2.**

- (i)
- ${\varphi}^{\diamond}=\varphi ,$
- (ii)
- ${E}^{\diamond}$ is $(\mathfrak{g},\mathcal{P})$-closed,
- (iii)
- ${\left({E}^{\diamond}\right)}^{\diamond}\subseteq {E}^{\diamond},$
- (iv)
- ${E}^{\diamond}\subseteq {F}^{\diamond},$ whenever $E\subseteq F,$
- (v)
- ${}^{\diamond}E\cup {F}^{\diamond}={(E\cup F)}^{\diamond},$
- (vi)
- ${(E\cap F)}^{\diamond}\subseteq {E}^{\diamond}\cap {F}^{\diamond}.$

**Theorem 3.**

**Lemma 1.**

**Theorem 4.**

**Corollary 1.**

**Definition 4.**

**Remark 3.**

**Theorem 5.**

- (i)
- $c{l}^{\diamond}\left(\varphi \right)=\varphi ,$
- (ii)
- $E\subseteq c{l}^{\diamond}\left(E\right),$
- (iii)
- $c{l}^{\diamond}\left(c{l}^{\diamond}\left(E\right)\right)=c{l}^{\diamond}\left(E\right),$
- (iv)
- $c{l}^{\diamond}\left(E\right)\subseteq c{l}^{\diamond}\left(F\right),$ whenever $E\subseteq F,$
- (v)
- $c{l}^{\diamond}\left(E\right)\cup c{l}^{\diamond}\left(F\right)=c{l}^{\diamond}(E\cup F).$

**Theorem 6.**

## 2. Methodology

## 3. Main Results

#### 3.1. Some Classes of $(\mathfrak{g},\mathcal{P})$-Open Sets

**Definition 5.**

- (i)
- When $E\subseteq c{l}^{\diamond}\left({i}_{\mathfrak{g}}E\right),$ E is named a $(\mathfrak{g},\mathcal{P})$-semi-open set.
- (ii)
- When $E\subseteq {i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left(E\right)\right),$ E is named a $(\mathfrak{g},\mathcal{P})$-pre-open set.
- (iii)
- When $E={i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left(E\right)\right),$ E is named a $(\mathfrak{g},\mathcal{P})$-regular open set.
- (iv)
- When $E\subseteq {c}_{\mathfrak{g}}\left({i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left(E\right)\right)\right),$ E is named a $(\mathfrak{g},\mathcal{P})$-β-open set.
- (v)
- When $E\subseteq {i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left({i}_{\mathfrak{g}}\left(E\right)\right)\right),$ E is named a $(\mathfrak{g},\mathcal{P})$-α-open set.

**Example 1.**

**Example 2.**

**Example 3.**

**Definition 6.**

**Theorem 7.**

- (i)
- Each $(\mathfrak{g},\mathcal{P})$-semi-open set is $\mathfrak{g}$-semi-open.
- (ii)
- Each $(\mathfrak{g},\mathcal{P})$-α-open set is $\mathfrak{g}$-α-open.
- (iii)
- Each $(\mathfrak{g},\mathcal{P})$-β-open set is $\mathfrak{g}$-β-open.
- (iv)
- Each $(\mathfrak{g},\mathcal{P})$-pre-open set is $\mathfrak{g}$-pre-open.

**Proof.**

**Theorem 8.**

- (i)
- E forms a $(\mathfrak{g},\mathcal{P})$-α-open set iff E is $(\mathfrak{g},\mathcal{P})$-semi-open as well as $(\mathfrak{g},\mathcal{P})$-pre-open.
- (ii)
- Considering E as $(\mathfrak{g},\mathcal{P})$-semi-open, E is $(\mathfrak{g},\mathcal{P})$-β-open.
- (iii)
- Considering E as $(\mathfrak{g},\mathcal{P})$-pre-open, E is $(\mathfrak{g},\mathcal{P})$-β-open.

**Proof.**

**Corollary 2.**

- (i)
- $\pi \cap \sigma =\alpha .$
- (ii)
- $\mathfrak{g}-\mathit{open}\subset \alpha \subset \sigma \subset \beta .$
- (iii)
- $\alpha \subset \pi \subset \beta .$

**Proposition 1.**

- (i)
- E is $(\mathfrak{g},\mathcal{P})$-regular open;
- (ii)
- E is $(\mathfrak{g},\mathcal{P})$-semi-closed as well as $(\mathfrak{g},\mathcal{P})$-open;
- (iii)
- E is $(\mathfrak{g},\mathcal{P})$-pre-open as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (iv)
- E is $(\mathfrak{g},\mathcal{P})$-α-open as well as $(\mathfrak{g},\mathcal{P})$-β-closed;
- (v)
- E is $(\mathfrak{g},\mathcal{P})$-α-open as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (vi)
- E is $(\mathfrak{g},\mathcal{P})$-open as well as $(\mathfrak{g},\mathcal{P})$-β-closed.

**Proof.**

**Remark 4.**

**Remark 5.**

**Example 4.**

- (i)
- E is $\mathfrak{g}$-semi-open, where ${c}_{\mathfrak{g}}\left({i}_{\mathfrak{g}}\right)=\mathbb{X}.$ However, E is not $(\mathfrak{g},\mathcal{P})$-semi-open, where $c{l}^{\diamond}\left({i}_{\mathfrak{g}}\right)=\{{x}_{1},{x}_{2}\}.$
- (ii)
- E is $(\mathfrak{g},\mathcal{P})$-β-open, where ${c}_{\mathfrak{g}}\left({i}_{\mathfrak{g}}\left({c}_{\mathfrak{g}}\left(E\right)\right)\right)=\mathbb{X}.$ However, E is not $(\mathfrak{g},\mathcal{P})$-semi-open.
- (iii)
- E is $(\mathfrak{g},\mathcal{P})$-β-open. However, E is not $(\mathfrak{g},\mathcal{P})$-pre-open, where ${i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left(E\right)\right)=\{{x}_{1},{x}_{2}\}$.

**Example 5.**

**Example 6.**

**Proposition 2.**

- (i)
- The countable union of $(\mathfrak{g},\mathcal{P})$-semi-open sets is $(\mathfrak{g},\mathcal{P})$-semi-open.
- (ii)
- The countable union of $(\mathfrak{g},\mathcal{P})$-pre-open sets is $(\mathfrak{g},\mathcal{P})$-pre-open.
- (iii)
- The countable union of $(\mathfrak{g},\mathcal{P})$-α-open sets is $(\mathfrak{g},\mathcal{P})$-α-open.
- (iv)
- The countable union of $(\mathfrak{g},\mathcal{P})$-β-open sets is $(\mathfrak{g},\mathcal{P})$-β-open.

**Proof.**

**Corollary 3.**

#### 3.2. Regular $(\mathfrak{g},\mathcal{P})$-Semi-Open and $(\mathfrak{g},\mathcal{P})$-Dense

**Definition 7.**

**Proposition 3.**

- (i)
- E is regular $(\mathfrak{g},\mathcal{P})$-semi-open;
- (ii)
- E is $(\mathfrak{g},\mathcal{P})$-semi-open as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (iii)
- E is $(\mathfrak{g},\mathcal{P})$-β-open as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (iv)
- E is $(\mathfrak{g},\mathcal{P})$-semi-open as well as $(\mathfrak{g},\mathcal{P})$-β-closed.

**Proof.**

**Remark 6.**

**Definition 8.**

**Remark 7.**

**Example 7.**

**Lemma 2.**

**Proof.**

**Theorem 9.**

- (i)
- $c{l}^{\diamond}\left(E\right)=c{l}^{\diamond}\left({i}_{\mathfrak{g}}E\right)\u27fa$ E is $(\mathfrak{g},\mathcal{P})$-semi-open.
- (ii)
- E is $(\mathfrak{g},\mathcal{P})$-semi-open ⟺ ∃ a $(\mathfrak{g},\mathcal{P})$-open set F satisfying $F\subseteq E\subseteq c{l}^{\diamond}\left(F\right).$
- (iii)
- For $E\subseteq G,$$G\subseteq c{l}^{\diamond}\left(E\right).$ Hence, G is $(\mathfrak{g},\mathcal{P})$-semi-open whenever E is $(\mathfrak{g},\mathcal{P})$-semi-open.
- (iv)
- Whenever E is $(\mathfrak{g},\mathcal{P})$-semi-open and F is $(\mathfrak{g},\mathcal{P})$-open, $E\cap F$ is $(\mathfrak{g},\mathcal{P})$-semi-open.

**Proof.**

**Theorem 10.**

**Proof.**

**Remark 8.**

**Example 8.**

**Theorem 11.**

**Proof.**

**Theorem 12.**

- (i)
- Whenever E is a $(\mathfrak{g},\mathcal{P})$-pre-closed set, $c{l}^{\diamond}\left({i}_{\mathfrak{g}}\left(E\right)\right)\subseteq E.$
- (ii)
- Whenever E is a $(\mathfrak{g},\mathcal{P})$-α-closed set, ${c}_{\mathfrak{g}}\left({i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left(E\right)\right)\right)\subseteq E.$
- (iii)
- Whenever E is a $(\mathfrak{g},\mathcal{P})$-β-closed set, ${i}_{\mathfrak{g}}\left(c{l}^{\diamond}\left({i}_{\mathfrak{g}}\left(E\right)\right)\right)\subseteq E.$

**Proof.**

**Theorem 13.**

**Proof.**

**Corollary 4.**

**Theorem 14.**

**Proof.**

**Corollary 5.**

**Proposition 4.**

- (i)
- Whenever $E\in \sigma $ and $F\in \alpha ,$$E\cap F\in \sigma .$
- (ii)
- Whenever $E\in \pi $ and $F\in \alpha ,$$E\cap F\in \pi .$
- (iii)
- Whenever $E,F\in \alpha ,$ $E\cap F\in \alpha .$

**Proof.**

## 4. Decomposition of $(\mathfrak{g},\mathcal{P})$-Continuity

**Definition 9.**

**Definition 10.**

**Theorem 15.**

- (i)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-α-continuous;
- (ii)
- $\forall \phantom{\rule{4pt}{0ex}}x\in \mathbb{X}$ and ${\mathtt{E}}^{\u2035}\in {\mathfrak{g}}^{\u2035}$ satisfy $\mathcal{U}\left(x\right)\in {\mathtt{E}}^{\u2035},$ and there exists $\mathtt{E}\in \alpha $ satisfying $x\in \mathtt{E}$ and $\mathcal{U}\left(\mathtt{E}\right)\subset {\mathtt{E}}^{\u2035};$
- (iii)
- ${\mathcal{U}}^{-1}\left({\mathtt{F}}^{\u2035}\right):{\mathtt{F}}^{\u2035}$ is $\mathfrak{g}$-closed and $(\mathfrak{g},\mathcal{P})$-closed.

**Proof.**

**Theorem 16.**

- (i)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-α-continuous iff $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-semi continuous as well as $(\mathfrak{g},\mathcal{P})$-pre continuous.
- (ii)
- Each $(\mathfrak{g},\mathcal{P})$-semi-continuous as well as each $(\mathfrak{g},\mathcal{P})$-pre-continuous set is $(\mathfrak{g},\mathcal{P})$-β-continuous.

**Proof.**

**Theorem 17.**

**Proof.**

**Theorem 18.**

**Proof.**

**Definition 11.**

**Remark 9.**

**Example 9.**

**Example 10.**

**Definition 12.**

**Remark 10.**

- (i)
- Each $\mathfrak{g}$-open function is $(\mathfrak{g},\mathcal{P})$-semi-open.
- (ii)
- Each $(\mathfrak{g},\mathcal{P})$-semi-open (respectively, $(\mathfrak{g},\mathcal{P})$-semi-closed) function is $\mathfrak{g}$-semi-open (respectively, $\mathfrak{g}$-semi-closed).

**Example 11.**

**Example 12.**

**Theorem 19.**

**Proof.**

**Theorem 20.**

**Proof.**

**Corollary 6.**

**Theorem 21.**

- (i)
- ${\mathcal{U}}^{-1}$ is $(\mathfrak{g},\mathcal{P})$-semi-continuous;
- (ii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-semi-open;
- (iii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-semi-closed.

**Proof.**

**Remark 11.**

**Example 13.**

**Example 14.**

**Definition 13.**

**Theorem 22.**

- (i)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-regular continuous;
- (ii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-pre-continuous as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (iii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-α-continuous as well as $(\mathfrak{g},\mathcal{P})$-semi-closed.

**Proof.**

**Corollary 7.**

- (i)
- $\mathcal{U}$ is regular $(\mathfrak{g},\mathcal{P})$-semi-continuous;
- (ii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-semi-continuous as well as $(\mathfrak{g},\mathcal{P})$-semi-closed;
- (iii)
- $\mathcal{U}$ is $(\mathfrak{g},\mathcal{P})$-β-continuous as well as $(\mathfrak{g},\mathcal{P})$-semi-closed.

**Proof.**

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Levine, N. Semi-open and semi-continuity in topological spaces. Am. Math. Mon.
**1963**, 70, 36–41. [Google Scholar] [CrossRef] - Al-Ghour, S.; Mansur, K. Between open sets and semi-open sets. Univ. Sci.
**2018**, 23, 9–20. [Google Scholar] [CrossRef] - Darwesh, H. A new type of semi-open sets and semi-continuity in topological spaces. Zan. J. Pur. Appl. Sci.
**2011**, 23, 82–94. [Google Scholar] - Srinivasa, V. On Semi-open sets and semi-separability. Glob. J. Inc.
**2013**, 13, 17–20. [Google Scholar] - Mashhour, A.; Abd El-Monsef, M.; El-Deeb, S. On precontinuous and weak precontinuous mappings. Proc. Math. Phys. Soc. Egypt
**1982**, 53, 47–53. [Google Scholar] - Abd El-Monswf, M.; El-Deeb, S.; Mahmoud, R. β-open sets and β-continuous mappings. Bull. Fac. Sci. Assiut Univ.
**1983**, 12, 77–90. [Google Scholar] - Najastad, O. On some classes of nearly open sets. Pac. J. Math.
**1965**, 15, 961–970. [Google Scholar] [CrossRef] - Andrijevic, D. On b-open sets. Mat. Vesn.
**1996**, 48, 59–64. [Google Scholar] - Császár, A. Generalized open sets. Acta Math. Hung.
**1997**, 75, 65–87. [Google Scholar] [CrossRef] - Aponte, E.; Subramanian, V.; Macias, J.; Krishnan, M. On semi-continuous and clisquish functions in generalized topological spaces. Axioms
**2023**, 12, 130. [Google Scholar] [CrossRef] - Korczak-Kubiak, E.; Loranty, A.; Pawlak, R.J. Baire generalized topological spaces, generalized metric spaces and infinite games. Acta Math. Hung.
**2013**, 140, 203–231. [Google Scholar] [CrossRef] - Császár, A. Generalized topology, generalized continuity. Acta Math. Hung.
**2002**, 96, 351–357. [Google Scholar] [CrossRef] - Császár, A. Remarks on quasi topologyies. Acta. Math. Hung.
**2008**, 119, 197–200. [Google Scholar] [CrossRef] - Császár, A. Separation axioms for generalized topologies. Acta Math. Hung.
**2004**, 104, 63–69. [Google Scholar] [CrossRef] - Ge, X.; Ge, Y. μ-Separations in generalized topological spaces. Appl. Math. J. Chin. Univ.
**2010**, 25, 243–252. [Google Scholar] [CrossRef] - Császár, A. Generalized open sets in generalized topologies. Acta Math. Hung.
**2005**, 106, 53–66. [Google Scholar] [CrossRef] - Császár, A. Extremally disconnected generalized topologies. Ann. Univ. Sci. Bp.
**2004**, 47, 91–96. [Google Scholar] - Császár, A. δ- and θ-modifications of generalized topologies. Acta. Math. Hung.
**2008**, 120, 275–279. [Google Scholar] [CrossRef] - Kuratowski, K. Topology, 1st ed.; Elsevier: Amsterdam, The Netherlands, 1966. [Google Scholar]
- Janković, D.; Hamlett, T. New topologies from old via ideals. Am. Math. Mon.
**1990**, 97, 295–310. [Google Scholar] [CrossRef] - Choquet, G. Sur les notions de filtre et de grille. Comptes Rendus Acad. Sci. Paris
**1947**, 224, 171–173. [Google Scholar] - Roy, B.; Mukherjee, M. On a typical topology induced by a grill. Soochow J. Math.
**2007**, 33, 771–786. [Google Scholar] - Al-Omari, A.; Noiri, T. On Ψ
_{G}-sets in grill topological spaces. Filomat**2011**, 25, 187–196. [Google Scholar] [CrossRef] - Al-Omari, A.; Noiri, T. On Ψ
_{*}-operator in ideal m-spaces. Bol. Soc. Paran. Math.**2012**, 30, 53–66. [Google Scholar] [CrossRef] - Al-Omari, A.; Noiri, T. On Ψ
_{G}-operator in grill topological spaces. An. Univ. Oradea Fasc. Mat.**2012**, 19, 187–196. [Google Scholar] - Acharjee, S.; Özkoç, M.; Issaka, F. Primal topological spaces. arXiv
**2022**. [Google Scholar] [CrossRef] - AL-Omari, A.; Acharjee, S.; Özkoç, M. A new operator of primal topological spaces. arXiv
**2022**. [Google Scholar] [CrossRef] - Mejías, L.; Vielma, J.; Guale, A.; Pineda, E. Primal topologies on finite-dimensional vector spaces induced by matrices. Int. J. Math. Sci.
**2023**. [Google Scholar] [CrossRef] - Al-Saadi, H.; Al-Malki, H. Generalized primal topological spaces. AIMS Math.
**2023**, 8, 24162–24175. [Google Scholar] [CrossRef]

**Figure 1.**Relationships between the $\mathfrak{g}$-open set and other types of open sets in $\mathbf{GPT}$ space.

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Al-Saadi, H.; Al-Malki, H.
Categories of Open Sets in Generalized Primal Topological Spaces. *Mathematics* **2024**, *12*, 207.
https://doi.org/10.3390/math12020207

**AMA Style**

Al-Saadi H, Al-Malki H.
Categories of Open Sets in Generalized Primal Topological Spaces. *Mathematics*. 2024; 12(2):207.
https://doi.org/10.3390/math12020207

**Chicago/Turabian Style**

Al-Saadi, Hanan, and Huda Al-Malki.
2024. "Categories of Open Sets in Generalized Primal Topological Spaces" *Mathematics* 12, no. 2: 207.
https://doi.org/10.3390/math12020207