Section 21: The Metric Topology (continued)

General properties (continued)

Metric spaces are Hausdorff.
A subspace of a metric space has the topology induced by the restriction of the space metric to the subspace.
An isometric imbedding of a metric space $X$ into a metric space $Y$ is a function $f:X\rightarrow Y$ such that $d_{Y}(f(a),f(b))=d_{X}(a,b)$ .
The countable product of metrizable spaces $(X_{i},d_{i})$ is metrizable in the product topology.
- $\rho(\mathbf{x},\mathbf{y})=\max\{d_{i}(x_{i},y_{i})\}_{i=1}^{n}$ and $D(\mathbf{x},\mathbf{y})=\sup\{\overline{d}_{i}(x_{i},y_{i})/i\}_{i\in\mathbb{Z}_{+}}$ are metrics for the product topology on $\prod_{i=1}^{n}(X_{i},d_{i})$ and $\prod_{i\in\mathbb{Z}_{+}}(X_{i},d_{i})$ , respectively.

Convergence in metric spaces

(The sequence lemma) Let $X$ be a topological space, then

in any topological space: if there is a sequence of points of $A\subset X$ converging to $x$ then $x\in\overline{A}$ .
in a metric space: if $x\in\overline{A}$ then there is a sequence of points of $A$ converging to $x$ .

Continuity in metric spaces

$\epsilon-\delta$ -definition of continuity: $f:(X,d_{X})\rightarrow(Y,d_{Y})$ is continuous at $x$ iff for every $\epsilon>0$ there is $\delta>0$ such that $d_{X}(x,y)<\delta$ implies $d_{Y}(f(x),f(y))<\epsilon$ .

Let $f:X\rightarrow Y$ , then

in any topological space: if $f$ is continuous at $x$ then for every $x_{n}\rightarrow x$ , $f(x_{n})\rightarrow f(x)$ .
in a metric space $X$ (and $Y$ is an arbitrary space): if for every $x_{n}\rightarrow x$ , $f(x_{n})\rightarrow f(x)$ , then $f$ is continuous at $x$ .

Uniform convergence

A sequence of functions $f_{n}:X\rightarrow(Y,d_{Y})$ converges uniformly to $f$ if for every $\epsilon>0$ there is $N\in\mathbb{Z}_{+}$ such that $d_{Y}(f_{n}(x),f(x))<\epsilon$ for all $n>N$ and $x\in X$ .

Uniformity of convergence depends not only on the topology of $Y$ but also on its metric.

(Uniform limit theorem) If $f_{n}:X\rightarrow(Y,d_{Y})$ are continuous and the sequence converges uniformly to $f$ then $f$ is continuous.

Moreover, in this case, for any $x_{n}\rightarrow x$ , $f_{n}(x_{n})\rightarrow f(x)$ .

Subpages

Section 21: The Metric Topology (continued)

Section 21: The Metric Topology (continued)

General properties (continued)

Convergence in metric spaces

Continuity in metric spaces

Uniform convergence