]> No Title

Metric Space

$\mathrm{\left[}\mathrm{\right]}$ Let X be any unstructured nonempty set.

$\mathrm{\left[}\mathrm{\right]}$ Let $\mathit{d}\mathrm{:}\mathrm{X}\mathrm{×}\mathrm{X}\mathrm{\to }\mathbb{R}$ be a function that assigns a real number to any pair of elements in the set X. Intuitively, it takes a pair of elements in the set X as the input and provides the distance between them (some real number) as the output. We could also say that $\mathit{d}$ takes an element from the set $\mathrm{X}\mathrm{×}\mathrm{X}$ as the input and produces a real number as the output.

$\mathrm{\left[}\mathrm{\right]}$ The tuple (X, d) is called a metric space if the following three conditions hold.

$\mathrm{•}$ Nonnegativity: $\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{\ge }\mathrm{0}\mathrm{\forall }\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{\in }\mathrm{X}$, and $\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{0}$ iff $\mathit{x}\mathrm{=}\mathit{y}\mathrm{.}$

$\mathrm{•}$ Symmetry: $\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{=}\mathit{d}\mathrm{\left(}\mathit{y}\mathrm{,}\mathrm{ }\mathit{x}\mathrm{\right)}\mathrm{\forall }\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{•}$ Triangular inequality: $\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{\le }\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{z}\mathrm{\right)}\mathrm{+}\mathit{d}\mathrm{\left(}\mathit{z}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{\forall }\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{,}$ $\mathit{z}\mathrm{\in }\mathrm{X}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ The function $\mathit{d}$ is called a metric on the set X.

$\mathrm{\left[}\mathrm{\right]}$ Unless necessary, we will denote the metric space as X.

$\mathrm{\left[}\mathrm{\right]}$ $\mathit{\epsilon }$neighborhood of $\mathit{x}$ in X: For any $\mathit{x}\mathrm{\in }\mathrm{X}$ and $\mathit{\epsilon }\mathrm{\in }{\mathbb{R}}_{\mathrm{+}\mathrm{+}}$, the $\mathit{\epsilon }$neighborhood of $\mathit{x}$ in $\mathit{X}$ is

$\mathrm{•}$ the set ${\mathit{N}}_{\mathit{\epsilon }\mathrm{,}\mathrm{X}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{\left\{}\mathit{y}\mathrm{\in }\mathrm{X}\mathrm{ }\mathrm{:}\mathrm{ }\mathit{d}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{ }\mathit{y}\mathrm{\right)}\mathrm{<}\mathit{\epsilon }\mathrm{\right\}}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Open subset of X: A subset $\mathit{S}$ of X is said to be open in X if

$\mathrm{•}$ for each $\mathit{x}\mathrm{\in }\mathit{S}\mathrm{,}$

$\mathrm{•}$ $\mathrm{\exists }\mathit{\epsilon }\mathrm{\in }{\mathbb{R}}_{\mathrm{+}\mathrm{+}}$ such that,

$\mathrm{•}{\mathit{N}}_{\mathit{\epsilon }\mathrm{,}\mathrm{X}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\subseteq }\mathit{S}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Closed subset of X: A subset $\mathit{S}$ of X is said to be closed in X if

$\mathrm{•}$ $\mathrm{X}\mathrm{\setminus }\mathit{S}$ is open in X.

$\mathrm{\left[}\mathrm{\right]}$ $\mathit{S}\mathrm{\subset }\mathrm{X}$ may be open, not open, closed, not closed, clopen, or neither open nor closed.

$\mathrm{\left[}\mathrm{\right]}$ Closure of a subset of X: Given any $\mathit{S}\mathrm{\subseteq }\mathit{X}$, the closure of $\mathit{S}$ in X is

$\mathrm{•}$ the smallest closed set $\mathit{C}{\mathit{l}}_{\mathit{X}}\mathrm{\left(}\mathit{S}\mathrm{\right)}$ such that

$\mathrm{•}\mathit{S}\mathrm{\subseteq }\mathit{C}{\mathit{l}}_{\mathit{X}}\mathrm{\left(}\mathit{S}\mathrm{\right)}$ .

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