On low dimensional KC-spaces (Q1946996)

A topological space \(X\) is said to be a KC-space if all compact subsets of \(X\) are closed, and is said to be weakly Hausdorff if every continuous map from a compact Hausdorff space into \(X\) has closed image in \(X\). It is well-known that the following implications hold for any topological space: Hausdorff \(\Rightarrow\) KC \(\Rightarrow\) Weakly Hausdorff. With these implications in mind, we are able to consider KC as an intermediary property between Hausdorff and weakly Hausdorff. It is also clear that such a property may be also considered as an intermediary separation axiom between \(T_2\) and \(T_1\). In the paper under review, the author shows that some specific, additional properties may turn a KC-space into a Hausdorff one. The main results of the paper are the following: {\parindent=6mm \begin{itemize}\item{}Theorem 1. Suppose \(X\) is a locally path connected KC-space containing no simple closed curve. Then, \(X\) is Hausdorff. \item{}Theorem 2. Suppose the KC-space \(X\) is generalized hereditarily unicoherent and suppose \(X\) is locally connected by continua. Then, \(X\) is Hausdorff. \end{itemize}} In the remainder of the paper, the author presents a number of counterexamples, showing that such additional hypotheses cannot be dropped. However, he points out at the end of the paper (motivated by his own theorems and counterexamples) that it is still an open question whether there is a contractible, \(1\)-dimensional, non-Hausdorff KC-space.