### $(\alpha ,\beta ,\theta ,\partial ,\mathcal{I})$-continuous mappings and their decomposition.

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Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open...

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces $(X,{\tau}_{X})$ and $(Y,{\tau}_{Y})$ are called T₁-complementary provided that there exists a bijection f: X → Y such that ${\tau}_{X}$ and ${f}^{-1}\left(U\right):U\in {\tau}_{Y}$ are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size ${2}^{}$ which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...

A hit-and-miss topology (${\tau}_{HM}$) is defined for the hyperspaces ${2}^{X}$, Cₙ(X) and Fₙ(X) of a continuum X. We study the relationship between ${\tau}_{HM}$ and the Vietoris topology and we find conditions on X for which these topologies are equivalent.

On the set $\mathbb{R}$ of real numbers we consider a poset ${\mathcal{P}}_{\tau}\left(\mathbb{R}\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq \mathbb{R}$, such that ${A}_{1}\supseteq {A}_{2}$ iff $\tau \left({A}_{1}\right)\subseteq \tau \left({A}_{2}\right)$. The poset has the minimal element $\tau \left(\mathbb{R}\right)$, the Euclidean topology, and the maximal element $\tau \left(\varnothing \right)$, the Sorgenfrey topology. We are interested when two topologies ${\tau}_{1}$ and ${\tau}_{2}$ (especially, for ${\tau}_{2}=\tau \left(\varnothing \right)$) from the poset define homeomorphic spaces $(\mathbb{R},{\tau}_{1})$ and $(\mathbb{R},{\tau}_{2})$. In particular, we prove that for a closed subset $A$ of $\mathbb{R}$ the space $(\mathbb{R},\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(\mathbb{R},\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common properties...

For a multivalued map $\varphi :Y\u22b8(X,\tau )$ between topological spaces, the upper semifinite topology $\mathcal{A}\left(\tau \right)$ on the power set $\mathcal{A}\left(X\right)=\{A\subset X:A\ne \varnothing \}$ is such that $\varphi $ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi :Y\to \left(\mathcal{A}\right(X),\mathcal{A}(\tau \left)\right)$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma $ on $\mathcal{A}\left(X\right)$, we consider a natural topology $\mathcal{R}\left(\Gamma \right)$ on $X$, constructed from $\Gamma $ satisfying $\mathcal{R}\left(\Gamma \right)=\tau $ if $\Gamma =\mathcal{A}\left(\tau \right)$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi :Y\u22b8(X,\mathcal{R}(\Gamma \left)\right)$...

We show a new theorem which is a sufficient condition for maximal resolvability of a topological space. We also discuss some relationships between various theorems about maximal resolvability.

We show that AC is equivalent to the assertion that every compact completely regular topology can be extended to a compact Tychonoff topology.