Talk:Totally disconnected space

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)[reply]

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.

David9550 (talk) 17:29, 26 March 2015 (UTC)[reply]

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)[reply]

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)[reply]