On the set $ℝ$ of real numbers we consider a poset ${𝒫}_{\tau }\left(ℝ\right)$ (by inclusion) of topologies $\tau \left(A\right)$, where $A\subseteq ℝ$, 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 $\tau \left(ℝ\right)$, 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 $(ℝ,{\tau }_1)$ and $(ℝ,{\tau }_2)$. In particular, we prove that for a closed subset $A$ of $ℝ$ the space $(ℝ,\tau (A\left)\right)$ is homeomorphic to the Sorgenfrey line $(ℝ,\tau (\varnothing \left)\right)$ iff $A$ is countable. We study also common properties...

We obtain a principal topology and some related results. We also give some hints of possible applications. Some mathematical systems are both lattice and topological space. We show that a topology defined on the any bounded lattice is definable in terms of uninorms. Also, we see that these topologies satisfy the condition of the principal topology. These topologies can not be metrizable except for the discrete metric case. We show an equivalence relation on the class of uninorms on a bounded lattice...

We observe the existence of a $\sigma$-compact, separable topological group $G$ and a countable topological group $H$ such that the tightness of $G$ is countable, but the tightness of $G\times H$ is equal to $𝔠$.

In this paper, the α waybelow relation, which is determined by O2-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for O2-convergence to be topological is obtained.

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

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le 2^{}$, a topological group G such that $G^{\gamma }$ is countably compact for all cardinals γ < α, but $G^{\alpha }$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has remained...

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.