Topology on a set
High school
A topology on a set is a choice of which subsets count as “open.” It describes which points are near which, without requiring a definition of “distance”.
Undergraduate
Consider a set $X$ and a collection $\mathcal{T}$ of subsets of $X$. The collection $\mathcal{T}$ is a topology on $X$ if
- $\emptyset$ and $X$ are in $\mathcal{T}$,
- every union of sets in $\mathcal{T}$ is in $\mathcal{T}$, and
- every intersection of finitely many sets in $\mathcal{T}$ is in $\mathcal{T}$.
Each set in a topology is defined as an open set.
Examples
Examples
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The open sets of $\reals$ (the unions of open intervals)
Undergraduate
The open sets of $\reals$ form a topology because unions and finite intersections of open sets are open.
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$\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$, on $X = \setdef{1, 2, 3}$
Undergraduate
The collection $\setdef{\emptyset, \setdef{1}, \setdef{2, 3}, \setdef{1, 2, 3}}$ is a topology on $\setdef{1, 2, 3}$, because it contains $\emptyset$ and $\setdef{1, 2, 3}$, and every union and intersection of its sets is in it.
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All subsets of a set $X$ (its power set)
Undergraduate
The power set of $X$ is a topology on $X$: the discrete topology, in which every set is open.
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$\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$, on $X = \setdef{1, 2}$
Undergraduate
The collection $\setdef{\emptyset, \setdef{1}, \setdef{1, 2}}$ is a topology on $\setdef{1, 2}$, because it contains $\emptyset$ and $\setdef{1, 2}$ and is closed under unions and intersections.
Non-examples
Things that may be mistaken for examples.
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The Borel sets of $\reals$
Graduate
Every set is a union of one-point sets, which are Borel, but not every set is Borel. So the Borel sets are not closed under arbitrary unions.
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The subsets of $\reals$ that are finite or have a finite complement
Graduate
The finite and cofinite subsets of $\reals$ do not form a topology, because they are not closed under arbitrary unions: the union of the one-point sets $\setdef{n}$, for $n \in \nats$, is $\nats$, which is neither finite nor cofinite.
Properties
- Every metric defines a topology, whose open sets are the unions of open balls.
- The complement of an open set need not be open. The complements of open sets are called closed.
Notes
“Topology” is also the name of the branch of mathematics that studies topologies and continuous maps.
Distinctions involving this term
- Topology vs. σ-algebra: Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.