The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

Abstract

A topologized semilattice $X$ is complete if each non-empty chain $C\subset X$ has $\inf C\in \bar{C}$ and $\sup C\in \bar{C}$. It is proved that for any complete subsemilattice $X$ of a functionally Hausdorff semitopological semilattice $Y$ the partial order $P=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$. This implies that for any continuous homomorphism $h:X\to Y$ from a compete topologized semilattice $X$ to a functionally Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$. The functional Hausdorffness of $Y$ in these two results can be replaced by the weaker separation axiom ${\vec{T}}_{2\delta }$, defined in this paper.

arXiv comment6 pages

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Cited by in corpus (6)

• Complete topologized posets and semilattices
• Characterizing categorically closed commutative semigroups
• On images of complete topologized subsemilattices in sequential semitopological semilattices
• The interplay between weak topologies on topological semilattices
• On the semigroup ${\mathbf{B}}_{\omega }^{{\mathcal{F}}_n}$ which is generated by the family ${\mathcal{F}}_n$ of finite bounded intervals of $\omega$
• The Lawson number of a semitopological semilattice