Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

Some specific extensions of topologycal spaces

Document Type : Original Paper

Authors
Seyedeh Somayeh Mousavi
Mehrdad Karavan jahromi
Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran
10.22055/jamm.2025.49808.2376
Abstract
In this paper, certain types of topological extensions are considered, in which the Hausdorff, regular, and Tychonoff properties are preserved, or the extension results in a Tychonoff (regular) space that is $\omega$-irresolvable. We introduce $\mathcal{C}$-simple and splitting extensions. These extensions preserve the first and second countability. Moreover, using a $\mathcal{C}$-simple extension, we show that every dense-in-itself Tychonoff space has a strongly $\omega$-irresolvable dense-in-itself Tychonoff extension. Within the classes of $T_1$, Hausdorff, regular, or Tychonoff spaces, maximal first-countable and maximal second-countable spaces are irresolvable. We also prove that a topological space is $\omega$-resolvable if and only if $\tau$ has a $\mathcal{C}$-simple extension. Furthermore, we study extensions that are submaximal and preserve key properties such as the Hausdorff condition, connectedness, being dense-in-itself, and extreme disconnectedness.
Keywords
Stable extension
$\mathcal{C}$-simple stable extension
Splitting extension
Resolvability
Submaximal space
Subjects
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  • Receive Date 31 May 2025
  • Revise Date 09 September 2025
  • Accept Date 02 October 2025