Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.Comment: 4 page

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$, $\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous

On operations on some classes of discontinuous maps

A map $f:Xightarrow Y$ between topological spaces is calledscatteredly continuous (pointwise discontinuous) if for eachnon-empty (closed) subspace $Asubset X$ the restriction $f|_{A}$has a point of continuity. We define a map $f:Xo Y$ to be weaklydiscontinuous if for every non-empty subspace $Asubset X$ the set$D(f|_A)$ of discontinuity points of the restriction $f|_A$ isnowhere dense in $A$.In this paper we consider the composition, Cartesian and diagonalproduct of weakly discontinuous, scatteredly continuous andpointwise discontinuous maps

On operations on some classes of discontinuous maps

A map $f:X\rightarrow Y$ between topological spaces is called scatteredly continuous (pointwise discontinuous) if for each non-empty (closed) subspace $A\subset X$ the restriction $f|_{A}$ has a point of continuity. We define a map $f:X\to Y$ to be weakly discontinuous if for every non-empty subspace $A\subset X$ the set $D(f|_A)$ of discontinuity points of the restriction $f|_A$ is nowhere dense in $A$.In this paper we consider the composition, Cartesian and diagonal product of weakly discontinuous, scatteredly continuous and pointwise discontinuous maps.<br /