We calculate the exact value of the color number of a periodic homeomorphism without fixed points on a finite connected graph.

In this article we study the comaximal graph ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$ of the ring $C\left(X\right)$. We have tried to associate the graph properties of ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$, the ring properties of $C\left(X\right)$ and the topological properties of $X$. Radius, girth, dominating number and clique number of the ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$ are investigated. We have shown that $2\le Rad{\Gamma }_{{}_2}^{\text{'}}C\left(X\right)\le 3$ and if $\left|X\right|>2$ then $\mathrm{girth}{\Gamma }_{{}_2}^{\text{'}}C\left(X\right)=3$. We give some topological properties of $X$ equivalent to graph properties of ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$. Finally we have proved that $X$ is an almost $P$-space which does not have isolated points if and only if $C\left(X\right)$ is an almost regular ring...

We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31/2}$-space. In [9] we showed that $C_p\left(X\right)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_p\left(X\right)$ has countable fan tightness and the Reznichenko property. 2....

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g:X\to X$ are continuous maps then $h_A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A\left(f\right)=h_A(f{|}_{{\cap }_{n\ge 0}{f}^n\left(X\right)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

If $X$ is a space that can be mapped onto a metric space by a one-to-one mapping, then $X$ is said to have a weaker metric topology. In this paper, we give characterizations of sequence-covering compact images and sequentially-quotient compact images of spaces with a weaker metric topology. The main results are that (1) $Y$ is a sequence-covering compact image of a space with a weaker metric topology if and only if $Y$ has a sequence ${\left\{{ℱ}_i\right\}}_{i\in \mathbb{N}}$ of point-finite $cs$-covers such that ${\bigcap }_{i\in \mathbb{N}}\mathrm{st}(y,{ℱ}_i)=\left\{y\right\}$ for each $y\in Y$. (2) $Y$ is a sequentially-quotient...