nLab quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff




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(quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff)


π:(X,τ X)(Y,τ Y) \pi \;\colon\; (X, \tau_X) \longrightarrow (Y, \tau_Y)

be a continuous function between topological spaces such that

  1. (X,τ)(X, \tau) is a compact Hausdorff topological space;

  2. π\pi is a surjection and τ Y\tau_Y is the corresponding quotient topology.

Then the following are equivalent

  1. (Y,τ Y)(Y, \tau_Y) is itself a Hausdorff topological space;

  2. π\pi is a closed map.


The implicaton ((Y,τ Y)Hausdorff)(πclosed)\left( (Y, \tau_Y)\, \text{Hausdorff} \right) \Rightarrow \left( \pi \, \text{closed} \right) follows since maps from compact spaces to Hausdorff spaces are closed. We need to show the converse.

Hence assume that π\pi is a closed map. We need to show that for every pair of distinct points y 1y 2Yy_1 \neq y_2 \in Y there exist open neighbourhoods U y 1,U y 2τ YU_{y_1}, U_{y_2} \in \tau_Y which are disjoint, U y 1U y 2=U_{y_1} \cap U_{y_2} = \emptyset.

First notice that the singleton subsets {x},{y}Y\{x\}, \{y\} \in Y are closed. This is because they are images of singleton subsets in XX, by surjectivity of ff, and because singletons in a Hausdorff space are closed, and because images under ff of closed subsets are closed, by the assumption that ff is a closed map.

It follows that the pre-images

C 1π 1({y 1})AAC 2π 1({y 2}). C_1 \coloneqq \pi^{-1}(\{y_1\}) \phantom{AA} C_2 \coloneqq \pi^{-1}(\{y_2\}) \,.

are closed subsets of XX.

Now since compact Hausdorff spaces are normal it follows that we may find disjoint open subset U 1,U 2τ XU_1, U_2 \in \tau_X such that

C 1U 1AAAC 2U 2. C_1 \subset U_1 \phantom{AAA} C_2 \subset U_2 \,.

Moreover, by this lemma we may find these U iU_i such that they are both saturated subsets. Therefore finally this lemma says that the images π(U i)\pi(U_i) are open in (Y,τ Y)(Y,\tau_Y). These are now clearly disjoint open neighbourhoods of y 1y_1 and y 2y_2.



Consider the function

[0,2π]/ S 1 2 t (cos(t),sin(t)) \array{ [0,2\pi]/\sim &\longrightarrow& S^1 \subset \mathbb{R}^2 \\ t &\mapsto& (cos(t), sin(t)) }

metric topology .

This is clearly a continuous function and a bijection on the underlying sets. Moreover, since continuous images of compact spaces are compact and since the closed interval [0,1][0,1] is compact we also obtain another proof that the circle is compact.

Hence since continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms the above map is in fact a homeomorphism

[0,2π]/S 1. [0,2\pi]/\sim \;\simeq\; S^1 \,.

Last revised on May 28, 2017 at 18:08:48. See the history of this page for a list of all contributions to it.