discreteDiscrete22008-03-26 17:57:322008-03-27 05:43:34Definitiondiscrete space% this is the default PlanetMath preamble. as your knowledge
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A topological space $S$ is said to be \emph{discrete} \underline{iff} it bears the discrete topology.\\
When $S$ is a subset of a topological space $\mathcal T$ it is said to discrete \underline{iff} any of the following two equivalent conditions is met:
\begin{itemize}
\item The subspace topology on $S$ \PMlinkescapetext{induced} by the topology on $\mathcal T$ is the discrete topology.
\item $\forall x\in S$, $\exists U\subset {\mathcal T}$ neighborhood of $x$, such that $U\cap S=\{x\}$.
\end{itemize}
If $S$ is discrete, then for all sequences $(x_i)_{i\in{\mathbb N}} \in S$ that converge to some $x\in S$, there exists $N_0\in\mathbb N$ such that $\forall i\ge N_0$, $x_i=x$. The converse holds when $S$ is first countable. Notice that when $S$ i$S$ is a subset of a metric space $\mathcal T$, $S$ is automatically metrizable hence first countable.