2025-06-29

1180: For Net with Directed Index Set on Subset of Topological Space That Converges on Space, Convergence Is on Closure of Subset

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description/proof of that for net with directed index set on subset of topological space that converges on space, convergence is on closure of subset

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 for any net with directed index set on any subset of any topological space that converges on the space, the convergence is on the closure of the subset.

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: Structured Description


Here is the rules of Structured Description.

Entities:
T: { the topological spaces }
S: T
D: { the directed index sets }
N: :DT, N(D)S
p: T
//

Statements:
N converges to p

pS.
//


2: Proof


Whole Strategy: Step 1: suppose that pS and find a contradiction.

Step 1:

Let us suppose that N converges to p.

Let us suppose that pS.

By a local characterization of closure: any point on any topological space is on the closure of any subset if and only if its every neighborhood intersects the subset, there would be a neighborhood of p, NpT, such that NpS=.

But there would be an index, j0D, such that N(j)Np for every jD such that j0j, but N(j)S, which would mean that NpS, a contradiction.

So, pS.


References


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