216: In Nest of Topological Subspaces, Connected-ness of Subspace Does Not Depend on Superspace
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A description/proof of that in nest of topological subspaces, connected-ness of subspace does not depend on superspace
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topological space
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Starting Context
Target Context
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The reader will have a description and a proof of the proposition that in any nest of topological subspaces, the connected-ness of any subspace does not depend on the superspace of which the subspace is regarded to be a subspace.
Orientation
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There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any topological space, , and any nest of topological subspaces, , such that , if is connected or disconnected as a subspace of , is connected or disconnected respectively as a subspace of ; if is connected or disconnected as a subspace of , is connected or disconnected respectively as a subspace of .
2: Proof
Suppose that is disconnected as a subspace of . , where is non-empty open on . By the proposition that in any nest of topological subspaces, the openness of any subset on any subspace does not depend on the superspace of which the subspace is regarded to be a subspace, is open with regarded to be a subspace of . So, is disconnected as a subspace of .
Suppose that is disconnected as a subspace of . Likewise, is disconnected as a subspace of .
As [ is disconnected as a subspace of ] [ is disconnected as a subspace of ], [ is not disconnected as a subspace of ] [ is not disconnected as a subspace of ], but as "not disconnected" is nothing but 'connected', [ is connected as a subspace of ] [ is connected as a subspace of ].
Likewise, [ is connected as a subspace of ] [ is connected as a subspace of ].
References
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