2022-10-09

365: Pair of Open Sets of Connected Topological Space Is Finite-Open-Sets-Sequence-Connected

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A description/proof of that pair of open sets of connected topological space is finite-open-sets-sequence-connected

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 any pair of open sets of any connected topological space is finite-open-sets-sequence-connected.

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 connected topological space, T, and any pair of open sets, U1,U2T, U1 and U2 are finite-open-sets-sequence-connected.


2: Proof


Take any open cover, Sc={Uα}, of T that includes U1 and U2, which is always possible because we can take the set of neighborhoods of all the point, with U1 and U2 added. Take the equivalence class, Se={Uβ}Sc, of the open cover such that each element of the equivalence class is finite-open-sets-sequence-connected with U1 via some elements of Sc, which is obviously an equivalence class.

If Se did not equal Sc, Sr:=ScSe={Uγ} would be nonempty. As SeSr=Sc is an open cover, (βUβ)(γUγ)=T, but as T is connected, (βUβ)(γUγ) because otherwise, T would be the disjoint union of open sets, βUβ and γUγ. So, there would be a point, pT, such that pβUβ and pγUγ, which means that pUβ for an β and pUγ for a γ, so, pUβUγ, which is a contradiction, because as Uγ would share a point with Uβ, Uγ should be finite-open-sets-sequence-connected with U1. So, Se equals Sc.

So, U2Se, and U1 and U2 are finite-open-sets-sequence-connected.


3: Note


Although not every connected topological space is path-connected, any pair of open sets of any connected topological space is connected by way of open sets.


References


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