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

$\mathrm{•}$ the largest open set $\mathit{I}{\mathit{n}}_{\mathit{X}}\mathrm{\left(}\mathit{S}\mathrm{\right)}$ such that

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

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

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

$\mathrm{\left[}\mathrm{\right]}$ Convergent sequence in X: A sequence $\mathrm{\left\{}{\mathit{x}}^{\mathit{m}}\mathrm{\right\}}\mathrm{\in }{\mathit{X}}^{\mathrm{\infty }}$ converges to ${\mathit{x}}^{\mathrm{*}}\mathrm{\in }\mathrm{X}$ 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

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

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

$\mathrm{\left[}\mathrm{\right]}$ Closed subset of X: A subset $\mathit{S}$ is closed in X if, and only if,

$\mathrm{•}$ any sequence all of whose terms are in $\mathit{S}$

$\mathrm{•}$ converges to a point in $\mathit{S}$, if it converges at all.

$\mathrm{\left[}\mathrm{\right]}$ Bounded subset of X: A subset $\mathit{S}$ is bounded in X if

$\mathrm{•}$ $\mathrm{\exists }\mathit{\epsilon }\mathrm{>}\mathrm{0}$ such that

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

$\mathrm{•}$ for some $\mathit{x}\mathrm{\in }\mathit{S}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Connected subset of X: A subset $\mathit{S}$ is connected in X iff

$\mathrm{•}$ $\mathit{S}$ can not be written as $\mathit{S}\mathrm{\equiv }{\mathit{S}}_{\mathrm{1}}\mathrm{\cup }{\mathit{S}}_{\mathrm{2}}$

$\mathrm{•}$ where ${\mathit{S}}_{\mathrm{1}}\mathrm{\cap }{\mathit{S}}_{\mathrm{2}}\mathrm{=}\mathrm{\varnothing }$

$\mathrm{•}$ and ${\mathit{S}}_{\mathrm{1}}\mathrm{,}$ ${\mathit{S}}_{\mathrm{2}}$ are open in $\mathit{S}\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Connected Metric Space X: X is connected iff the only clopen subsets of X are X and $\mathrm{\varnothing }\mathrm{.}$

$\mathrm{\left[}\mathrm{\right]}$ Dense subset of X: A subset $\mathit{Y}$ is said to be dense in X if

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

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