Re: quotient topology

On Thu, 8 May 2008, [ISO-8859-1] Mariano Suárez-Alvarez wrote:
On May 8, 4:46 pm, berwald.f...@xxxxxxxxxxxxxx wrote:

Let X be topological space, R an equivalence relation on X and
X/R be the set whose elements are all the R-equivalence classes.

1. X/R is a T1 space iff every R-equivalence class closed in X.

2. Suppose X is a compact Hausdorff space.
X/R is Hausdorff iff the projection p:X--->X/R is closed.

Can anyone give me suggestions?
What have you tried doing in order to prove this?

The not trivial part is when p is closed,
to show that X/R is Hausdorff.

Since X is normal T1 and normality is preserved by
closed continuous surjections and T1 is preserved by
closed surjections, X/R is normal T1, hence Hausdorff.
.

Relevant Pages

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