2022-03-13

43: Local Criterion for Openness

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description/proof of local criterion for openness

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 local criterion for openness.

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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(S\): \(\subseteq T\)
//

Statements:
\(S \in \{\text{ the open subsets of } T\}\).
\(\iff\)
\(\forall p \in S (\exists U_p \subseteq T (U_p \in \{\text{ the open subsets of } T\} \land p \in U_p \subseteq S))\).
//


2: Natural Language Description


For any topological space, \(T\), any subset, \(S \subseteq T\), is open if and only if for any point, \(p \in S\), there is an open neighborhood, \(U_p\), of \(p\) such that \(p \in U_p \subseteq S\).


3: Proof


Let us suppose that \(U_p\) exists.

\(S\) is the union of such all the \(U_p\) s, because any point in \(S\) belongs to the union and any point in the union belongs to \(S\), so, as an union of open sets, \(S\) is an open set.

Let us suppose that \(S\) is open.

\(S\) is a \(U_p\).


References


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