Talk:Totally disconnected space
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Contradiction
[edit]1) definition contradicts summary; it describes a much stronger condition. specifically, it conflates connected _components_ with connected subspaces.
2) is there a totally disconneted but without base of clopen ? an example ? — Preceding unsigned comment added by 2.136.29.148 (talk) 07:51, 27 April 2012 (UTC)
- 1) According to the definition of connected components, the definition in the summary is equvalent to the definition in the main text. You may be thinking of connected quasi-components, which can be larger than connected components.
- 2) Yes: the latter condition (a base of clopen sets) is equivalent to zero-dimensionality, so any totally disconnected space which is not zero-dimensional provides a counterexample.
- I think people generally agree that the empty set is *not* connected because its number of connected components is zero (a connected space should have exactly one connected component). Maybe the first few sentences should be edited to reflect this? — Preceding unsigned comment added by 72.33.2.59 (talk) 21:49, 21 February 2017 (UTC)
Question
[edit]3) Is totally disconnected equivalent to every pair of points a,b in X can be separated by an open decomposition X=U\cupdot V, a in U, bin V?
Such an equivalent characterization could then be useful to be mentioned in the article.
Freeze S (talk) 13:50, 8 October 2020 (UTC)