Existence of non-constant continuous functions

Question

Under what circumstances is there at least one non-constant continuous function from a topological space $X$ to a topological space $Y$? Assume that $X$ and $Y$ each have at least two points. If $X$ is disconnected, separated by $A$ and $B$, then any function with one value on $A$ and another on $B$ is continuous. If $X$ is connected, then the image of $X$ under a continuous function must lie within a connected component of $Y$. Therefore, to avoid triviality, assume that $X$ and $Y$ are both connected.

The only theorem I've encountered of this nature is Urysohn's lemma, which proves such a function exists if $X$ is a $T_4$ space and $Y$ has a path-connected component with more than one point. This is of course a rather strong condition.

It's obvious that if $X$ is convex in $\mathbb{R}$ and $Y$ is totally path disconnected, then there is no such function.

Otherwise, I haven't a clue. I'm particularly curious about what happens if $X$ and/or $Y$ is required to be homogeneous or bihomogeneous, and/or if $Y$ is required to be uniform.

Answer 1

This paper might be of interest:

• Horst Herrlich: Wann sind alle stetigen Abbildungen in Y konstant? Mathematische Zeitschrift, Volume 90, Number 2, 152-154 Also freely available at GDZ.

The content of the paper:

Urysohn [5] asked whether for every regular space $X$ (having at least two points) there is a non-constant continuous map from $X$ do the space $Y$ of real numbers. This question was negatively answered by Hewitt [2], Novak [2] and Van Est-Freudenthal [1]. The methods used by these authors (which go back to Tychonoff [4]) let us show relatively easy the following result:

Theorem Let $Y$ be a topological space. The following conditions are equivalent:
(a) $Y$ is a $T_1$-space,
(b) there exists a regular space $X$ (having at least two points), such that every continuous map from $X$ to $Y$ is constant.

[SNIP] Proof of the above theorem. [SNIP]

Remark. The above results has a trivial analogue:
Let $X$ be a topological space. The following conditions are equivalent: (a) $X$ is connected,
(b) there is a regular space $Y$ (having at least 2 points), such that every continuous map from $X$ to $Y$ is constant.

Answer 2

I doubt that a complete and simple characterization exists. One common obstruction though (generalizing a bit what you said) is if $X$ is connected and $Y$ is completely disconnected.