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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

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