description/proof of that metric space Is \(T_1\) topological space
Topics
About: metric space
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 \(T_1\) topological space.
Target Context
- The reader will have a description and a proof of the proposition that any metric space with the topology induced by the metric is a \(T_1\) topological space.
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:
\(M\): \(\in \{\text{ the metric spaces }\}\), with the topology induced by the metric
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Statements:
\(M \in \{\text{ the } T_1 \text{ topological spaces }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(M\) satisfies the condition to be \(T_1\).
Step 1:
Let \(m_1, m_2 \in M\) be any such that \(m_1 \neq m_2\).
\(0 \lt dist (m_1, m_2)\), by the definition of metric.
Let us take \(B_{m_1, dist (m_1, m_2) / 2} \subseteq M\), which is an open neighborhood of \(m_1\), by Note for the definition of topology induced by metric.
\(m_2 \notin B_{m_1, dist (m_1, m_2) / 2}\), because \(dist (m_1, m_2) / 2 \le dist (m_1, m_2)\).
So, \(M\) is a \(T_1\) topological space.
