Summary

Section Summary

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

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If $(X, d)$ is a metric space and $a \in X,$ then an open ball centered at $a$ is a set of the form

$B (a, δ) = {x \in X ∣ d (x, a) < δ}$

for some positive number $δ .$
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A subset $N$ of a metric space $(X, d)$ is s neighborhood of a point $a \in N$ if there is a positive real number $δ$ such that $B (a, δ) \subseteq N .$
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An important property of open balls is that every open ball is a neighborhood of each of its points. This is our first step toward defining the concept of open sets that will form the foundation for topological spaces.
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A function $f$ from a metric space $(X, d_{X})$ to a metric space $(Y, d_{Y})$ is continuous at $a \in X$ if $f^{- 1} (N)$ is a neighborhood of $a$ in $X$ for any neighborhood $N$ of $f (a)$ in $Y .$