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Definition 1. A family $\mathcal{B}$ of non-empty open sets in a topological space will be called $\pi$-base (or pseudo-base) if every non-empty open set contains at least one member of $\mathcal{B}$. …

In a private communication, Professor Laszlo Zsilinszky mentioned to me the article "An example involving Baire spaces" of H. E. White Jr, in that article it is shown that: Theorem. If the Continuum …

On $(\mathbb{R}, \tau)$ the euclidean space of real numbers, we define a new topology by letting $\tau^{*}=\{X\subseteq \mathbb{R}: X=\emptyset \hspace{0.1cm}\mbox{or}\hspace{0.1cm}\mathbb{R}\setminus …

Studying the article "Games that involve set theory or topology" of Marion Scheepers, I found the following result Theorem 46 Let $\{(X_{i}, \tau_{i}) : i\in I \}$ be a family of topological spaces. …

Let $X$ be a non-empty topological space. The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. In the $n$-th inning Player I choo …

As Taras Banakh noted, there is a theorem for Tychonoff powers, Theorem Let $\kappa\geq \omega$. If $X^{\omega}$ is Baire, then $X^{\kappa}$ is Baire, where the powers are considered in the Tychono …