]> No Title

$\mathrm{\left[}\mathrm{\right]}$ Separable Metric Space X: X is separable if $\mathrm{•}$ $\mathrm{\exists }\mathit{Y}\mathrm{\subseteq }\mathit{X}$ such that

$\mathrm{•}\mathit{C}{\mathit{l}}_{\mathit{X}}\mathrm{\left(}\mathit{Y}\mathrm{\right)}\mathrm{=}\mathit{X}$

$\mathrm{•}$ and $\mathit{Y}$ is countable.

$\mathrm{\left[}\mathrm{\right]}$ Cover of $\mathit{S}\mathrm{\subseteq }\mathrm{X}$: A collection ${O}$ of subsets of X is a cover of $\mathit{S}$ in X if $\mathrm{•}\mathit{S}\mathrm{\subseteq }\mathrm{\cup }{O}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Open cover of $\mathit{S}\mathrm{\subseteq }\mathrm{X}$: A collection ${O}$ of subsets of X is an open cover of $\mathit{S}$ in X if $\mathrm{•}\mathit{S}\mathrm{\subseteq }\mathrm{\cup }{O}$

$\mathrm{•}$ and each member of ${O}$

$\mathrm{•}$ is open in X.

$\mathrm{\left[}\mathrm{\right]}$ Compact subset of X: A subset $\mathit{S}$ is compact in X if $\mathrm{•}$ every finite and infinite open cover of $\mathit{S}$ in X

$\mathrm{•}$ has a finite subset

$\mathrm{•}$ that also covers $\mathit{S}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Totally bounded subset of X: A subset $\mathit{S}$ is totally bounded in X if $\mathrm{•}$ for any $\mathit{\epsilon }\mathrm{>}\mathrm{0}$

$\mathrm{•}$ $\mathrm{\exists }$ a finite subset $\mathit{T}$ of $\mathit{S}$ such that

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

$\mathrm{\left[}\mathrm{\right]}$ Cauchy sequence in X: A sequence $\mathrm{\left\{}{\mathit{x}}^{\mathit{m}}\mathrm{\right\}}\mathrm{\in }\mathit{x}\mathrm{\infty }$ is said to be cauchy if $\mathrm{•}$ for each $\mathit{\epsilon }\mathrm{>}\mathrm{0}$

$\mathrm{•}$ $\mathrm{\exists }$ a real number $\mathit{M}\mathrm{\left(}\mathit{\epsilon }\mathrm{\right)}$ such that

$\mathrm{•}\mathit{d}\mathrm{\left(}{\mathit{x}}^{\mathit{k}}\mathrm{,}\mathrm{ }{\mathit{x}}^{\mathit{l}}\mathrm{\right)}\mathrm{<}\mathit{\epsilon }$

$\mathrm{•}\mathrm{\forall }\mathit{k}\mathrm{,}$ $\mathit{l}\mathrm{\ge }\mathit{M}\mathrm{\left(}\mathit{\epsilon }\mathrm{\right)}$ .

$\mathrm{\left[}\mathrm{\right]}$ Complete Metric Space X: A metric space is complete if every cauchy sequence in X $\mathrm{•}$ converges to a point in X.

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