Exercise 18, § 9

A Hausdorff space $X$ is said to be absolutely closed if it has the following property: for every homeomorphism $f$ of $X$ onto a subspace of a Hausdorff space $X'$, $f(X)$ is closed in $X'$. Every compact space is absolutely closed.

a) Show that the following conditions are equivalent:

(AF') The space $X$ is absolutely closed.

(AF'') Every filter base on $X$ which consists of open sets has at least one cluster point.

(AF''') Every family of closed subsets of $X$ whose intersection is empty contains a finite subfamily such that the interiors of the subsets in this subfamily have an empty intersection.

(AF''') Every open covering of $X$ contains a finite subfamily whose closures cover $X$.

b) Let X be a Hausdorff space and U a dense open subset of X. Suppose that every filter base on U which consists of open sets has at least one cluster point in X. Show that under these conditions X is absolutely closed. In particular, if A is an open subset of an absolutely closed space X, then $\overline{A}$ is absolutely closed.

c) Let X be an absolutely closed space and let $f$ be a continuous mapping of X into a Hausdorff space $X'$. Show that $f(X)$ is an absolutely closed subspace of $X'$.

d) Show that the product of two absolutely closed spaces is absolutely closed (argue as in Proposition 17 of no. 10).