Ivan Shishkin, Birch Grove

Topology on a set

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7/27/2026, 12:48:38 PM · Ancient Tree

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1A topology on a [[Set|set]] $X$ is a collection $\tau$ of subsets of $X$ called the [[Open set|open sets]], such that:
21) $\emptyset\in \tau$ and $X\in \tau$;
21. The [[Empty set|empty set]] and the whole space are open: $\emptyset\in \tau$ and $X\in \tau$.
32) $\tau$ is closed under arbitrary unions: if $\{U_{i}\}_{i\in I}\subseteq \tau$, then $\bigcup_{i \in I} U_i \in \tau$;
32. [[Arbitrary union|Arbitrary unions]] of open sets are open: if $\{U_{i}\}_{i\in I}$ is any family of sets in $\tau$, then $\bigcup_{i \in I} U_i \in \tau$.
43) $\tau$ is closed under finite intersections: if $U,V\in\tau$, then $U\cap V\in \tau$.
43. Finite [[Intersection of sets|intersections]] of open sets are open: if $U_1, \ldots, U_n \in \tau$, then $U_1 \cap \cdots \cap U_n \in \tau$.
5
6##### Remarks
7- A set $X$ together with a topology $\tau$ on $X$ is called a [[Topological space|topological space]].

Revision 555

7/22/2026, 12:43:28 PM · Ancient Tree

Concept edited

Compare with revision 385No text changes
1A topology on a [[Set|set]] $X$ is a collection $\tau$ of subsets of $X$ called the [[Open set|open sets]], such that:
21) $\emptyset\in \tau$ and $X\in \tau$;
32) $\tau$ is closed under arbitrary unions: if $\{U_{i}\}_{i\in I}\subseteq \tau$, then $\bigcup_{i \in I} U_i \in \tau$;
43) $\tau$ is closed under finite intersections: if $U,V\in\tau$, then $U\cap V\in \tau$.

Revision 385

7/10/2026, 1:35:29 PM · Ancient Tree

Concept edited

Compare with revision 384No text changes
1A topology on a [[Set|set]] $X$ is a collection $\tau$ of subsets of $X$ called the [[Open set|open sets]], such that:
21) $\emptyset\in \tau$ and $X\in \tau$;
32) $\tau$ is closed under arbitrary unions: if $\{U_{i}\}_{i\in I}\subseteq \tau$, then $\bigcup_{i \in I} U_i \in \tau$;
43) $\tau$ is closed under finite intersections: if $U,V\in\tau$, then $U\cap V\in \tau$.

Revision 384

7/10/2026, 1:34:20 PM · Ancient Tree

Concept created

A topology on a [[Set|set]] $X$ is a collection $\tau$ of subsets of $X$ called the [[Open set|open sets]], such that:
1) $\emptyset\in \tau$ and $X\in \tau$;
2) $\tau$ is closed under arbitrary unions: if $\{U_{i}\}_{i\in I}\subseteq \tau$, then $\bigcup_{i \in I} U_i \in \tau$;
3) $\tau$ is closed under finite intersections: if $U,V\in\tau$, then $U\cap V\in \tau$.