description/proof of local criterion for openness
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of topological space.
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 \(U_p\).
