2023-02-26

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

Topics


About: topological space

The table of contents of this article


Starting Context



Target Context


  • 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


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Description


For any topological space, T, and any nest of topological subspaces, T1,T2, such that T2T1T, if T2 is connected or disconnected as a subspace of T1, T2 is connected or disconnected respectively as a subspace of T; if T2 is connected or disconnected as a subspace of T, T2 is connected or disconnected respectively as a subspace of T1.


2: Proof


Suppose that T2 is disconnected as a subspace of T1. T2=U1U2, U1U2= where Ui is non-empty open on T2. 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, Ui is open with T2 regarded to be a subspace of T. So, T2 is disconnected as a subspace of T.

Suppose that T2 is disconnected as a subspace of T. Likewise, T2 is disconnected as a subspace of T1.

As [T2 is disconnected as a subspace of T1] [T2 is disconnected as a subspace of T], [T2 is not disconnected as a subspace of T] [T2 is not disconnected as a subspace of T1], but as "not disconnected" is nothing but 'connected', [T2 is connected as a subspace of T] [T2 is connected as a subspace of T1].

Likewise, [T2 is connected as a subspace of T1] [T2 is connected as a subspace of T].


References


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