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
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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, , and any pair of open sets, , and are finite-open-sets-sequence-connected.
2: Proof
Take any open cover, , of that includes and , which is always possible because we can take the set of neighborhoods of all the point, with and added. Take the equivalence class, , of the open cover such that each element of the equivalence class is finite-open-sets-sequence-connected with via some elements of , which is obviously an equivalence class.
If did not equal , would be nonempty. As is an open cover, , but as is connected, because otherwise, would be the disjoint union of open sets, and . So, there would be a point, , such that and , which means that for an and for a , so, , which is a contradiction, because as would share a point with , should be finite-open-sets-sequence-connected with . So, equals .
So, , and and 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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