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discrete
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(Definition)
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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:
- The subspace topology on $S$ induced by the topology on $\mathcal T$ is the discrete topology.
- $\forall x\in S$ $\exists U\subset {\mathcal T}$ neighborhood of $x$ such that $U\cap S=\{x\}$
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.
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"discrete" is owned by lalberti.
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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 103 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 1997 times total.
Classification:
| AMS MSC: | 54A05 (General topology :: Generalities :: Topological spaces and generalizations ) |
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Pending Errata and Addenda
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