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Filter Basis Convergent Sequence Neighborhood Metric Space Set Subspace Topology Converse Theorem Topological Space Sequence Discrete Metrizable First Countable

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Discrete

A topological space is said to be discrete iff it bears the discrete topology.
When is a subset of a topological space $\mathcal T$ it is said to discrete iff any of the following two equivalent conditions is met:

The subspace topology on induced by the topology on $\mathcal T$ is the discrete topology.
$\forall x\in S$ , $\exists U\subset {\mathcal T}$ neighborhood of , such that $U\cap S=\{x\}$ .

If

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 $\forall i\ge N_0$ ,

. The converse holds when

is first countable. Notice that when

i

is a subset of a metric space $\mathcal T$ ,

is automatically metrizable hence first countable.

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