Compactification of cut-point spaces

arXiv1909.10646MaRDI QIDQ6325849

David Sumner Lipham

Publication date: 23 September 2019

Abstract: We show that if $X$ is a separable locally compact Hausdorff connected space with fewer than $mathfrakc$ non-cut points, then $X$ embeds into a dendrite $Dsubseteqmathbb{R}^2$, and the set of non-cut points of $X$ is a nowhere dense $G_{d}elta$-set. We then prove a Tychonoff cut-point space $X$ is weakly orderable if and only if $$ is an irreducible continuum. Finally, we show every separable metrizable cut-point space densely embeds into a reducible continuum with no cut points. By contrast, there is a Tychonoff cut-point space each of whose compactifications has the same cut point. The example raises some questions about persistent cut points in Tychonoff spaces.

Mathematics Subject Classification ID

Continua and generalizations (54F15) Topological characterizations of particular spaces (54F65)

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