Summary

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Section Summary

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

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A metric on a space $X$ is a function that measures distance between elements in the space. More formally, a metric on a space $X$ is a function $d : X \times X \to R^{+} \cup {0}$ such that
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  $d (x, y) \geq 0$ for all $x, y \in X,$
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  $d (x, y) = 0$ if and only if $x = y$ in $X,$
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  $d (x, y) = d (y, x)$ for all $x, y \in X,$ and
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  $d (x, y) \leq d (x, z) + d (z, y)$ for all $x, y, z \in X .$
A metric space is any space combined with a metric defined on that space.
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The Euclidean, taxicab, and max metric are all metrics on $R^{n},$ so they all provide ways to measure distances between points in $R^{n} .$ These metric are different in how they define the distances.
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  The Euclidean metric is the standard metric that we have used through our mathematical careers. For elements $x = (x_{1}, x_{2}, \dots, x_{n})$ and $y = (y_{1}, y_{2}, \dots, y_{n})$ in $R^{n},$ the Euclidean metric $d_{E}$ is defined as
  
  $\begin{aligned} d_{E} (x, y) & = \sqrt{(x_{1} - y_{1})^{2} + (x_{2} - y_{2})^{2} + \dots (x_{n} - y_{n})^{2}} \\ = \sqrt{\sum_{i = 1}^{n} (x_{i} - y_{i})^{2}} . \end{aligned}$
  
  With this metric, the unit circle in $R^{2}$ (the set of points a distance $1$ from the origin) is the standard unit circle we know from Euclidean geometry.
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  The taxicab metric $d_{T}$ is defined as
  
  $\begin{aligned} d_{T} (x, y) & = | x_{1} - y_{1} | + | x_{2} - y_{2} | + \dots + | x_{n} - y_{n} | \\ = \sum_{i = 1}^{n} | x_{i} - y_{i} | . \end{aligned}$
  
  The unit circle in $R^{2}$ using the taxicab metric is the square with vertices $(1, 0),$ $(0, 1),$ $(- 1, 0),$ and $(0, - 1)$ when viewed in Euclidean geometry.
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  The max metric $d_{M}$ is defined by
  
  $\begin{aligned} d_{M} (x, y) & = max {| x_{1} - y_{1} |, | x_{2} - y_{2} |, | x_{3} - y_{3} |, \dots, | x_{n} - y_{n} |} \\ = max_{1 \leq i \leq n} {| x_{i} - y_{i} |} . \end{aligned}$
  
  Under the max metric, the unit circle in $R^{2}$ is the square with vertices $(1, 1),$ $(- 1, 1),$ $(- 1, - 1),$ and $(1, - 1)$ when viewed in Euclidean geometry.

Topology: An Inquiry-Based Approach

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