description/proof of that for \(1\)-dimensional Euclidean topological space, set of upper bounded open intervals and lower bounded open intervals is subbasis
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of subbasis of topological space.
Target Context
- The reader will have a description and a proof of the proposition that for the \(1\)-dimensional Euclidean topological space, the set of the upper bounded open intervals and the lower bounded open intervals is a subbasis.
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:
\(\mathbb{R}\): \(= \text{ the Euclidean topological space }\)
\(S\): \(= \{(- \infty, r) \subseteq \mathbb{R} \vert r \in \mathbb{R}\} \cup \{(r, \infty) \subseteq \mathbb{R} \vert r \in \mathbb{R}\}\)
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Statements:
\(S \in \{\text{ the subbases for } \mathbb{R}\}\)
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2: Proof
Whole Strategy: Step 1: see that \(S\) satisfies the conditions to be a subbasis.
Step 1:
The intersection of each finite subset of \(S\) is open, as a finite intersection of open subsets.
Let \(r \in \mathbb{R}\) be any.
Let \(N_r \subseteq \mathbb{R}\) be any neighborhood of \(r\).
There is a \(B_{r, \epsilon} \subseteq \mathbb{R}\) such that \(r \in B_{r, \epsilon} \subseteq N_r\).
\(B_{r, \epsilon} = (- \infty, r + \epsilon) \cap (r - \epsilon, \infty)\).
So, \(r \in (- \infty, r + \epsilon) \cap (r - \epsilon, \infty) \subseteq N_r\).
So, the set of the intersections of the finite subsets of \(S\) is a basis for \(\mathbb{R}\).
So, \(S\) is a subbasis for \(\mathbb{R}\).
