description/proof of that for topological space and point, intersection of finite number of neighborhoods of point is neighborhood of point
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of neighborhood of point on topological space.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and its any point, the intersection of any finite number of neighborhoods of the point is a neighborhood of the point.
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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(t\): \(\in T\)
\(\{N_{t, 1}, ..., N_{t, n}\}\): \(N_{t, n} \in \{\text{ the neighborhoods of } t \text{ on } T\}\)
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Statements:
\(\cap_{j \in \{1, ..., n\}} N_{t, j} \in \{\text{ the neighborhoods of } t \text{ on } T\}\)
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2: Proof
Whole Strategy: Step 1: let any open neighborhood of \(t\) contained in \(N_{t, j}\) be \(U_{t, j}\); Step 2: see that \(\cap_{j \in \{1, ..., n\}} U_{t, j}\) is an open neighborhood of \(t\) and \(t \in \cap_{j \in \{1, ..., n\}} U_{t, j} \subseteq \cap_{j \in \{1, ..., n\}} N_{t, j}\).
Step 1:
For each \(j \in \{1, ..., n\}\), there is an open neighborhood of \(t\) contained in \(N_{t, j}\), \(U_{t, j}\), by the definition of a definition of neighborhood of point on topological space.
Step 2:
\(\cap_{j \in \{1, ..., n\}} U_{t, j}\) is an open neighborhood of \(t\), as the intersection of the finite number of open subsets.
\(\cap_{j \in \{1, ..., n\}} U_{t, j} \subseteq \cap_{j \in \{1, ..., n\}} N_{t, j}\), because \(U_{t, j} \subseteq N_{t, j}\).
So, \(t \in \cap_{j \in \{1, ..., n\}} U_{t, j} \subseteq \cap_{j \in \{1, ..., n\}} N_{t, j}\) where \(\cap_{j \in \{1, ..., n\}} U_{t, j}\) is open.
So, \(\cap_{j \in \{1, ..., n\}} N_{t, j}\) is a neighborhood of \(t\).
