Quotient topology
The quotient topology is the canonical way of defining a topology on a Algebraic quotient,
as defined by an Equivalence relation or projection.
Let
Further characterisations
Universal property
For every topological space
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Proof
First we will prove that the quotient topology as characterised above satisfies the universal property. Let
be a topological space, ( π , T π ) be a surjective function, and π : π β π be endowed with the quotient topology π . Let T π be some topological space, and let ( π , T π ) be a function. If π : π β π is continuous, then so is the composition π of continuous functions. Now suppose π π is continuous, and let π π : π β π . Then π β T π whence ( π π ) β 1 π = π β 1 π β 1 π β T π . Thus π β 1 β T π is continuous. Therefore π is continuous iff π is continuous. π π Now let
be a topology on T β² satisfying the universal property. In particular, let π and ( π , T π ) = ( π , T π ) . Then since π = i d π : π β¦ π is continuous so is π π = π , wherefore π is finer than T β² Now let T π and ( π , T π ) = ( π , T β² ) . Since π = i d π is continuous, so too is i d π . But i d π β‘ π = π is the finest topology for which T π is continuous, so π . T π = T β²
Further terminology
- An equivalence relation is called a Closed equivalence relation iff it is closed regarded as a subset of
π Γ π
Properties
- From the universal property, a function
π : π β π π : π β π π : π β π - A function
π : π β π π π : π β π
Spaces constructed as quotients
- Unit circle topology as defined by
[ 0 , 1 ] 0 βΌ 1 - MΓΆbius strip, Klein bottle, and other shapes constructed using a Fundamental polygon
- Projective space
Footnotes
-
where
π π π / βΌ -
2020, Topology: A categorical approach, pp. 28β29 β©