description/proof of that topological space induced by metric is Hausdorff
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of Hausdorff topological space.
Target Context
- The reader will have a description and a proof of the proposition that the topological space induced by any metric is Hausdorff.
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:
\(T\): \(\in \{\text{ the topological spaces }\}\), induced by any metric, \(dist: T \times T \to \mathbb{R}\)
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Statements:
\(T \in \{\text{ the Hausdorff topological spaces }\}\)
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2: Proof
Whole Strategy: Step 1: for each \(t, t' \in T\) such that \(t \neq t'\), take the open neighborhood of \(t\), \(B_{t, dist (t, t') / 2}\), and the open neighborhood of \(t'\), \(B_{t', dist (t, t') / 2}\),, and see that \(B_{t, dist (t, t') / 2} \cap B_{t', dist (t, t') / 2} = \emptyset\).
Step 1:
Let \(t, t' \in T\) be any such that \(t \neq t'\).
\(0 \lt dist (t, t')\).
Let us take the open neighborhood of \(t\), \(B_{t, dist (t, t') / 2}\), and the open neighborhood of \(t'\), \(B_{t', dist (t, t') / 2}\),.
Let us see that \(B_{t, dist (t, t') / 2} \cap B_{t', dist (t, t') / 2} = \emptyset\).
Let \(u \in B_{t, dist (t, t') / 2}\) be any.
\(dist (t, t') \le dist (t, u) + dist (u, t')\), so, \(dist (t, t') - dist (t, u) \le dist (u, t')\), but \(dist (t, t') / 2 = dist (t, t') - dist (t, t') / 2 \lt dist (t, t') - dist (t, u)\), so, \(dist (t, t') / 2 \lt dist (u, t')\), which means that \(u \notin B_{t', dist (t, t') / 2}\).
So, \(B_{t, dist (t, t') / 2} \cap B_{t', dist (t, t') / 2} = \emptyset\).
