Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
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Section 3.2 Metrizability
“With a metric you can really go to town, otherwise it is just abstract nonsense.”
―Jennifer Tour Chayes
Definition 3.2.1 .
A metric on a set \(X\) is a function \(d: X \times X \to \R\) satisfying the following for all \(x,y,z \in X\text{:}\)
\(d(x,y) \geq 0\text{,}\) and \(d(x,y) = 0\) if and only if \(x = y\text{.}\)
\(d(x,y) = d(y,x)\text{.}\)
\(d(x,z) \leq d(x,y) + d(y,z)\text{.}\)
The pair \((X,d)\) is a metric space, and for any \(x,y \in X\) the real number \(d(x,y)\) is the distance from \(x\) to \(y\text{.}\)
Definition 3.2.2 .
Let \(d\) be a metric on a set \(X\text{.}\) For any \(p \in X\) and \(r \in \R\) with \(r > 0\text{,}\) the ball of radius \(r\) centered at \(p\) is \(B_{d(p,r)} := \{q \in X | d(p,q) < r\}\text{.}\)
The metric basis on \(X\) induced by \(d\) is the collection \(\cB_d := \{B_{d(p,r)} | p \in X \text{ and }r > 0\}\text{.}\)
The metric topology on \(X\) induced by \(d\) is the topology \(\cT_d := \cT (\cB_d)\) generated by the metric basis.
Proposition 3.2.3 .
Let \((X,d)\) be a metric space. The metric basis on \(X\) is a basis, and hence the metric topology is a topology.
Theorem 3.2.4 .
Let \((X,d)\) be a metric space, let \(X\) have the metric topology, and let \(X \times X\) have the product topology. Then the metric function \(d: X \times X \to (\R,\cT_{\text{Eucl}})\) is continuous.
Example 3.2.5 .
Examples
Definition 3.2.6 .
A topological space \((X,\cT_X)\) is metrizable if there is a metric \(d\) on \(X\) such that \(\cT_X = \cT_d\) (where \(\cT_d\) is the metric topology on \(X\) induced by \(d\) ).
Theorem 3.2.7 .
Metrizability is a homeomorphism invariant.
Interactions with constructions and continuous functions:.
Proposition 3.2.8 .
If \(X\) is a metrizable space and \(Y\) is a subspace of \(X\text{,}\) then \(Y\) is metrizable.
If \(X\) and \(Y\) are metrizable spaces, then the product space \(X \times Y\) is metrizable.
Interactions with homeomorphism invariants:.
Theorem 3.2.10 .
Metrizable spaces are \(T_2\text{.}\)
