definition of topology induced by metric
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of metric.
- The reader knows a definition of topology.
Target Context
- The reader will have a definition of topology induced by metric.
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:
\( T\): \(\in \{\text{ the metric spaces }\}\)
\(*O\): \(\in \{\text{ the topologies of } T\}\)
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Conditions:
\(\forall S \subseteq T (S \in O \iff (\forall p \in S (\exists \epsilon \in \mathbb{R} (0 \lt \epsilon \land (B_{p, \epsilon} \subseteq S)))))\).
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2: Natural Language Description
For any metric space, \(T\), the topology, \(O\), such that for any subset, \(S \subseteq T\), \(S\) is open if and only if for each \(p \in S\), there is a positive, \(\epsilon \in \mathbb{R}\), such that \(B_{p, \epsilon} \subseteq S\)
3: Note
It is indeed a topology: \(\emptyset \in O\); \(T \in O\); for any possibly uncountable number of open subsets, \(\{U_\alpha \vert \alpha \in A\}\), and \(U := \cup_{\alpha \in A} U_\alpha\), for any \(p \in U\), \(B_{p, \epsilon_\alpha} \subseteq U_\alpha\) for any fixed \(\alpha\), and \(B_{p, \epsilon_\alpha} \subseteq U_\alpha \subseteq U\); for any finite number of open subsets, \(U_1, ..., U_k\), and \(U := \cap_{j = 1, .., k} U_k\), for any \(p \in U\), \(B_{p, \epsilon_j} \subseteq U_j\) for each \(j\), and for \(\epsilon := \min \{\epsilon_j\}\), \(0 \lt \epsilon\) and \(B_{p, \epsilon} \subseteq U\).
For each \(p \in T\) and each \(\epsilon\), \(B_{p, \epsilon} \in O\), because for any \(p' \in B_{p, \epsilon}\), \(dist (p, p') \le \epsilon\), so, for \(B_{p', \epsilon - dist (p, p')}\), for each \(p'' \in B_{p', \epsilon - dist (p, p')}\), \(dist (p'', p') \lt \epsilon - dist (p, p')\), but \(dist (p'', p) \le dist (p'', p') + dist (p', p) \lt \epsilon - dist (p, p') + dist (p', p) = \epsilon\), which means that \(B_{p', \epsilon - dist (p, p')} \subseteq B_{p, \epsilon}\).
