Topological spaces · Definition
Topology
Before you start you need to know
A topology abstracts the essential feature of the open sets of a metric space and lets us speak of nearness, limits and continuity without distances.
Definition
A topology on a set \(X\) is a family \(\mathcal{T}\) of subsets of \(X\), whose members \(U\) are called open sets, containing \(\varnothing\) and \(X\) and closed under arbitrary unions and finite intersections.
In a metric space the open sets satisfy three properties: the empty set and the whole space are open, every union of open sets is open, and the intersection of two open sets is open. The definition of a topology takes exactly those properties as axioms, so any family satisfying them counts as «the open sets» of \(X\), whether or not it comes from a distance.
Asking for the intersection of two open sets suffices: by induction one gets any finite number. Asking for infinite intersections would be too much, since \(\bigcap_n(-1/n,1/n)=\{0\}\) is not open in \(\mathbb{R}\). A guiding example is \(X=\{a,b,c\}\) with \(\mathcal{T}=\{\varnothing,X,\{a\},\{a,b\}\}\): one checks by hand that every union and intersection lands in \(\mathcal{T}\) again.
An intersection of topologies on \(X\) is a topology, but a union in general is not (new unions and intersections must be added). A common mistake is to think that any family containing \(\varnothing\) and \(X\) is already a topology, or that open sets must be «intervals»: on a finite set they can be almost anything.
With this axiom we will define the topological space, and afterwards closed sets, bases, interiors and continuity, all in terms of open sets.
Think about it
Why does the definition ask only for finite intersections?
Because infinite intersections would already fail in \(\mathbb{R}\): \(\bigcap_n(-1/n,1/n)=\{0\}\), which is not open.
Is \(\{\varnothing,\{a\},\{b\},X\}\) a topology on \(X=\{a,b\}\)?
Yes: it lists all four subsets of \(X\), so it is the family of all subsets, and unions and intersections of subsets are subsets.
Exercises 10
- In \(X=\{a,b,c\}\), which of the following families is a topology?
- How many distinct topologies exist on the set \(\{1,2\}\)?
- Each interval \((-1/n,1/n)\) is open in \(\mathbb{R}\). What is the intersection of all of them, \(n\ge1\)?
- On \(\mathbb{N}\) the open sets are \(\varnothing\) and the sets with finite complement. Which set is open?
- The family of all subsets of \(X=\{1,2,3,4\}\) is a topology. How many open sets does it have?
- In \(X=\{a,b,c\}\) let \(\mathcal{T}_1=\{\varnothing,X,\{a\}\}\) and \(\mathcal{T}_2=\{\varnothing,X,\{b\}\}\). The union \(\mathcal{T}_1\cup\mathcal{T}_2\) is not a topology. Which set is missing?
- If \(\mathcal{T}_1\) and \(\mathcal{T}_2\) are topologies on \(X\), which of these families is always a topology?
- How many open sets does the smallest topology on \(\{1,2,3\}\) containing \(\{1,2\}\) and \(\{2,3\}\) have?
- Which of the following families of subsets of \(\mathbb{R}\) is a topology?
- Let \(X\) be infinite with open sets \(\varnothing\) and the sets with finite complement. If \(U\) and \(V\) are nonempty open sets, what happens?
What comes next
References
- James R. Munkres, Topology, 2nd edition, Pearson, 2000. §12
- Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 3