TL;DR
This paper investigates the properties of low-dimensional KC-spaces, highlighting conditions under which they are Hausdorff and providing examples of non-Hausdorff cases with interesting connectedness properties.
Contribution
It offers new insights into the structure of low-dimensional KC-spaces, including examples and conditions affecting their separation properties.
Findings
Certain low-dimensional KC-spaces are Hausdorff under specific conditions.
Examples of compact, connected, non-Hausdorff KC-spaces are constructed.
Nested intersections of compact connected subsets can be disconnected in these spaces.
Abstract
The KC property, a separation axiom between weakly Hausdorff and Hausdorff, requires compact subsets to be closed. Various assumptions involving local conditions, dimension, connectivity, and homotopy show certain KC-spaces are in fact Hausdorff. Several low dimensional examples of compact, connected, non-Hausdorff KC-spaces are exhibited in which the nested intersection of compact connected subsets fails to be connected.
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