Metric Topology#
Definition: Open Ball
Let \((M, d)\) be a metric space and let \(p \in M\).
The open ball of radius \(r \gt 0\) around \(p\) is the set of all elements in \(M\) whose distance from \(p\) is less than \(r\).
Notation
Theorem: The Metric Topology
Let \((M,d)\) be a metric space.
The collection of all open balls in \(M\) forms a base for a topological space \((M, \tau_d)\).
Proof
TODO
Definition: Metric Topology
The topology \(\tau_d\) is known as the metric topology induced on \(M\) by \(d\).
Definition: Equivalent Metrics
Let \(M\) be a set.
We say that two metrics are equivalent if they induce the same metric topology on \(M\).
Definition: Metrizable Space
A topological space \((X, \tau)\) is metrizable if there exists a metric \(d\) on \(X\) such that \(\tau\) is the metric topology induced by \(d\) on \(X\).
Theorem: Open Sets in the Metric Topology
Let \((M,d)\) be a metric space and let \(\tau\) be the metric topology induced on \(M\) by \(d\).
A subset \(U \subseteq M\) is open in \(\tau\) if and only if for each \(u \in U\) there exists an open ball \(B_{\varepsilon} (u)\) such that \(B_{\varepsilon}(u) \subseteq U\).
Proof
This follows directly from topology generation.
Theorem: Continuity in Metric Spaces
Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces with their respective metric topologies.
A function \(f: X \to Y\) is continuous at \(p \in X\) if and only if for each open ball \(B_\varepsilon(f(p))\) around \(f(p)\) there exists some open ball \(B_\delta(p)\) around \(p\) such that if \(x\) is inside \(B_\delta(p)\), then \(f(x)\) is inside \(B_\varepsilon(f(p))\).
Proof
TODO
Theorem: Compactness via Limits
Let \((X, d)\) be a metric space with its metric topology \(\tau_d\).
A subset \(S \subseteq X\) is compact if and only if every sequence of points in \(S\) has a convergent (in terms of \(\tau_d\)) subsequence with a limit in \(S\).
Proof
TODO