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%%% Primary Title: discrete
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%%% Owner: lalberti
%%% Author(s): lalberti
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\begin{document}

 A \htmladdnormallink{topological space}{http://planetmath.org/encyclopedia/TopologicalSpace.html} $S$ is said to be \emph{discrete} \underline{iff} it bears the \htmladdnormallink{discrete topology}{http://planetmath.org/encyclopedia/DiscreteTopologicalSpace.html}.\\
When $S$ is a \htmladdnormallink{subset}{http://planetmath.org/encyclopedia/ProperSubset2.html} of a topological space $\mathcal T$ it is said to discrete \underline{iff} any of the following two \htmladdnormallink{equivalent}{http://planetmath.org/encyclopedia/Equivalent4.html} conditions is met:
\begin{itemize}
\item The \htmladdnormallink{subspace topology}{http://planetmath.org/encyclopedia/SubspaceTopology.html} on $S$ induced by the topology on $\mathcal T$ is the discrete topology.
\item $\forall x\in S$, $\exists U\subset {\mathcal T}$ \htmladdnormallink{neighborhood}{http://planetmath.org/encyclopedia/DeletedNeighborhood.html} of $x$, such that $U\cap S=\{x\}$.
\end{itemize}
If $S$ is discrete, then for all \htmladdnormallink{sequences}{http://planetmath.org/encyclopedia/Sequence.html} $(x_i)_{i\in{\mathbb N}} \in S$ that \htmladdnormallink{converge}{http://planetmath.org/encyclopedia/LimitPoint2.html} to some $x\in S$, there exists $N_0\in\mathbb N$ such that $\forall i\ge N_0$, $x_i=x$. The \htmladdnormallink{converse}{http://planetmath.org/encyclopedia/ConverseTheorem2.html} holds when $S$ is \htmladdnormallink{first countable}{http://planetmath.org/encyclopedia/FirstCountable.html}. Notice that when $S$ i$S$ is a subset of a \htmladdnormallink{metric space}{http://planetmath.org/encyclopedia/MetricTopology.html} $\mathcal T$, $S$ is automatically \htmladdnormallink{metrizable}{http://planetmath.org/encyclopedia/Metrizable.html} hence first countable.

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