description/proof of that Euclidean metric space is locally compact
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of Euclidean metric.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of locally compact topological space.
- The reader admits the Heine-Borel theorem.
Target Context
- The reader will have a description and a proof of the proposition that any Euclidean metric space is locally compact.
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:
\(\mathbb{R}^d\): \(= \text{ the Euclidean metric space }\), with the induced topology
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Statements:
\(\mathbb{R}^d \in \{\text{ the locally compact topological spaces }\}\)
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2: Note
A metric space is not necessarily local compact, because the Heine-Borel theorem does not necessarily hold for a general metric space.
3: Proof
Whole Strategy: Step 1: let \(t \in \mathbb{R}^d\) be any and let \(N_t \subseteq \mathbb{R}^d\) be any neighborhood of \(t\), and take any \(B_{t, \epsilon'}\) such that \(B_{t, \epsilon'} \subseteq N_t\) and take any \(B_{t, \epsilon}\) such that \(\overline{B_{t, \epsilon}} \subseteq B_{t, \epsilon'}\); Step 2: see that \(\overline{B_{t, \epsilon}}\) is a compact neighborhood of \(t\).
Step 1:
Let \(t \in \mathbb{R}^d\) be any.
Let \(N_t \subseteq \mathbb{R}^d\) be any neighborhood of \(t\).
There is an \(\epsilon'\)-'open ball' around \(t\), \(B_{t, \epsilon'} \subseteq \mathbb{R}^d\), such that \(B_{t, \epsilon'} \subseteq N_t\).
For any \(\epsilon \in \mathbb{R}\) such that \(\epsilon \lt \epsilon'\), \(\overline{B_{t, \epsilon}} \subseteq B_{t, \epsilon'}\), because \(t' \in \overline{B_{t, \epsilon}}\) implies that for each \(\epsilon'' \in \mathbb{R}\), \(B_{t', \epsilon''} \cap B_{t, \epsilon} \neq \emptyset\), so, there is a \(t'' \in B_{t', \epsilon''} \cap B_{t, \epsilon}\), and \(dist (t', t) \le dist (t', t'') + dist (t'', t) \lt \epsilon'' + \epsilon\), but as \(\epsilon''\) is arbitrary, \(dist (t', t) \le \epsilon \lt \epsilon'\).
\(\overline{B_{t, \epsilon}} \subseteq N_t\).
Step 2:
\(\overline{B_{t, \epsilon}}\) is bounded and closed on \(\mathbb{R}^d\).
By the Heine-Borel theorem, \(\overline{B_{t, \epsilon}}\) is a compact subset of \(\mathbb{R}^d\).
\(\overline{B_{t, \epsilon}}\) contains the open neighborhood of \(t\), \(B_{t, \epsilon}\).
So, \(\overline{B_{t, \epsilon}}\) is a compact neighborhood of \(t\) that is contained in \(N_t\).
