Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $\varrho (x,y)=W\left(A_{xy}\right)$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...

We establish the topological relationship between compact Hausdorff spaces X and Y equivalent to the existence of a bound-2 isomorphism of the sup norm Banach spaces C(X) and C(Y).

Let $L\left(X\right)$ be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space $X$, where, by identifying each $f$ with the epi-graph $epi\left(f\right)$, $L\left(X\right)$ is regarded the subspace of the space ${Cld}_F^{*}(X\times ℝ)$ of all closed sets in $X\times ℝ$ with the Fell topology. Let $$LSC\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\cap ℝ\ne \varnothing ,f\left(X\right)\subset (-\infty ,\infty ]\}\text{and}{LSC}_B\left(X\right)=\{f\in L\left(X\right)\mid f\left(X\right)\text{is}\text{a}\text{bounded}\text{subset}\text{of}ℝ\}.$$ We show that $L\left(X\right)$ is homeomorphic to the Hilbert cube $Q={[-1,1]}^{\mathbb{N}}$ if and only if $X$ is second countable, locally compact and infinite. In this case, it is proved that $(L\left(X\right),LSC\left(X\right),{LSC}_B\left(X\right))$ is homeomorphic to $(ConeQ,Q\times (0,1),\Sigma \times (0,1\left)\right)$ (resp. $(Q,s,\Sigma )$) if $X$ is compact (resp. $X$ is non-compact), where $ConeQ=(Q\times 𝐈)/(Q\times \{1\left\}\right)$ is the cone over...