Quotient of zero-dimensional hausdorff space

Question

I've read (in joy of cats) that every topological space is a regular quotient of a zero-dimensional hausdorff space. So far, I could not find a proof. Do you know one, or a reference?

Answer 1

I will assume that by regular quotient you mean existence of regular epimorphism (in Top), i.e. of a quotient map. I hope I did not overlook something. (The proof seems to be relatively easy.)

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Let us say that $X$ is a prime space if it has only one non-isolated point.

If $a\in X$, then the prime factor $X_a$ of $X$ at $a$ is the space which has the same neighborhoods of $a$ as $X$ and all points other than $a$ are isolated.

It is easy to see that:

• Every topological space is a quotient of a sum of its prime factors.

• A prime factor is either discrete or prime space.

Discrete spaces are obviously zero-dimensional, Hausdorff prime space are zero-dimensional and a sum of Hausdorff zero-dimensional spaces is zero-dimensional. Therefore it is sufficient to show that every prime space can be obtained as a quotient of a sum of Hausdorff zero-dimensional spaces.

Suppose that a prime space $X$ with the accumulation point $a$ is non-Hausdorff. Let $C$ be an intersection of all neighborhoods of $a$. Then the space $X$ can be obtained as a quotient of a sum of:

• the space $X\setminus C\cup\{a\}$ (which is a prime Hausdorff space);

• several copies of Sierpinski space, one for each point of $C\setminus \{a\}$.

Now our problem reduces to obtaining Sierpinski space as a quotient of a Hausdorff prime space, which is easy. (There are many possibilities, e.g. you can obtain is as a quotient of $\{0\}\cup\{\frac1n;n\in\mathbb N\}$ with the topology inherited from $\mathbb R$.)

I use the term prime factor and prime space in the same way as in the following papers. (Although I do not claim that Franklin and Rajagopalan were the first ones to use these notions.)

• Franklin S.P., Rajagopalan M., On subsequential spaces, Topology Appl. 35 (1990), 1--19.

• Zhou J.-Y., On subspaces of pseudo-radial spaces, Comment.Math.Univ.Carolinae 34,3 (1993), 583--586