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[parent] discrete (Definition)

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

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.



"discrete" is owned by lalberti.
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See Also: discrete space


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Cross-references: metrizable, metric space, first countable, converse, converge, sequences, neighborhood, subspace topology, equivalent, subset, discrete topology, topological space
There are 73 references to this entry.

This is version 3 of discrete, born on 2008-03-26, modified 2008-03-27.
Object id is 10445, canonical name is Discrete2.
Accessed 256 times total.

Classification:
AMS MSC: 54A05 (General topology :: Generalities :: Topological spaces and generalizations )
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