Discrete topology
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Revision 843
8/2/2026, 3:47:24 PM · Sequoia
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Revision 841
8/2/2026, 3:45:10 PM · Sequoia
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The discrete topology on a set $X$ is the [[topology|topology]] such that the open sets are the subsets of $X$, hence $\tau:=\mathcal{P}(X)$.1
The discrete topology on a set $X$ is the [[topology|topology]] such that the [[Open set|open sets]] are the subsets of $X$, hence $\tau:=\mathcal{P}(X)$.2
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##### Remarks4
- In particular, all singletons $\{x\}$ are both open and closed since $\{x\}\subset X$ and $\{x\}$ is the complement of $X\backslash\{x\}\subset X$.4
- In particular, all singletons $\{x\}$ are both open and [[Closed set|closed]] since $\{x\}\subset X$ and $\{x\}$ is the complement of $X\backslash\{x\}\subset X$.Revision 839
8/2/2026, 3:43:32 PM · Sequoia
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This older revision predates detailed metadata tracking.
Revision 838
8/2/2026, 3:43:30 PM · Sequoia
Concept edited
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1
The discrete topology on a set $X$ is the [[topology|topology]] $\tau=\mathcal{P}(X)$.1
The discrete topology on a set $X$ is the [[topology|topology]] such that the open sets are the subsets of $X$, hence $\tau:=\mathcal{P}(X)$.2
3
##### Remarks4
- In particular, all singletons $\{x\}$ are both open and closed.4
- In particular, all singletons $\{x\}$ are both open and closed since $\{x\}\subset X$ and $\{x\}$ is the complement of $X\backslash\{x\}\subset X$.Revision 822
8/1/2026, 7:13:37 PM · Ancient Tree
Concept created
Recorded titleDiscrete topology
Recorded typeDefinition
The discrete topology on a set $X$ is the [[topology|topology]] $\tau=\mathcal{P}(X)$.
##### Remarks
- In particular, all singletons $\{x\}$ are both open and closed.