Topological space
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Revision 664
7/27/2026, 12:47:06 PM · Ancient Tree
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A topological space is a [[Set|set]] $X$ together with a [[topology|topology]] $\tau$, that is, a collection of subsets of $X$, called the [[Open set|open sets]], satisfying the following axioms:2
1. The [[Empty set|empty set]] and the whole space are open: $\emptyset\in \tau$ and $X\in \tau$.3
2. [[Arbitrary union|Arbitrary unions]] of open sets are open: if $\{U_{i}\}_{i\in I}$ is any family of sets in $\tau$, then $\bigcup_{i \in I} U_i \in \tau$.4
3. Finite [[Intersection of sets|intersections]] of open sets are open: if $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$.Revision 663
7/27/2026, 12:46:39 PM · Ancient Tree
Concept edited
Compare with revision 1795 changed lines
1
A topological space is...1
A topological space is a [[Set|set]] $X$ together with a [[topology|topology]] $\tau$, that is, a collection of subsets of $X$, called the [[Open set|open sets]], satisfying the following axioms:2
1. The [[Empty set|empty set]] and the whole space are open: $\emptyset\in \tau$ and $X\in \tau$.3
2. [[Arbitrary union|Arbitrary unions]] of open sets are open: if $\{U_{i}\}_{i\in I}$ is any family of sets in $\tau$, then $\bigcup_{i \in I} U_i \in \tau$.4
3. Finite [[Intersection of sets|intersections]] of open sets are open: if $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$.Revision 179
7/1/2026, 1:37:07 PM · Ancient Tree
Concept created
A topological space is...