Summary

Section Summary

Important ideas that we discussed in this section include the following.

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A subset $A$ of a metric space $(X, d)$ is a metric space, called a subspace, by using the metric $d |_{A \times A}$ on $A .$
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If $X$ is a metric space and $A$ is a subspace of $X,$ a subset $O_{A}$ of $A$ is open in $A$ if and only if $O_{A} = X \cap O$ for some open set $O$ in $X .$ A subset $C_{A}$ of $A$ is closed in $A$ if $C_{A} = A \cap O_{A}$ for open set $O_{A}$ in $A .$ Alternatively, a set $C_{A}$ is closed in $A$ if $C_{A} = A \cap C$ for some closed set $C$ in $X .$
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Let $(X_{i}, d_{i})$ be metric spaces for $i$ from $1$ to some positive integer $n .$ The product metric space $(X, d)$ is the Cartesian product

$X = X_{1} \times X_{2} \times \dots \times X_{n} = \prod_{i = 1}^{n} X_{i}$

with metric $d$ defined by

$d (x, y) = \sqrt{\sum_{i = 1}^{n} d_{i} (x_{i}, y_{i})^{2}}$

when $x = (x_{1}, x_{2}, \dots, x_{n})$ and $y = (y_{1}, y_{2}, \dots, y_{n})$ are in $X .$