Difference between revisions of "A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces"
From Maths
(Created page with "__TOC__ ==Statement== Let {{Top.|X|J}} be a topological space, then{{rITTMJML}}: * {{Top.|X|J}} is {{link|disconnected|topology}} (ie: not {{link|connected|topology}}) {{i...") |
(No difference)
|
Latest revision as of 23:15, 30 September 2016
Contents
Statement
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, then[1]:
- [ilmath](X,\mathcal{ J })[/ilmath] is disconnected (ie: not connected) if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
Proof
Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
This proof has been marked as an page requiring an easy proof
The message provided is:
Exercise in Lee's top. manifolds, didn't take me very long to do, note that A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements is a useful "precursor" theorem
This proof has been marked as an page requiring an easy proof
See also
References