Existence of non-constant continuous functions

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; MR32:3029, Zbl 0131.20402.

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.

Sketch of the construction in the proof of this theorem: (I have omitted many details and also the proofs that these space do have the required properties.)

Definition of a space $Q$. First we start with some given space $Y$.

• The spaces $R_i$ for $i=1,2$ and points $r_i\in R_i$ are constructed in such way that every continuous map from $R_i$ to $Y$ is constant on some neighborhood of $R_i$.
• A space $T=R_1\times R_2\setminus \{(r_1,r_2)\}$. This space has the property, that for every continuous map $f$ from $T$ to $Y$ there exist neighborhoods $U_i$ of $r_i$ such that $f$ is constant on $U_1\times U_2 - \{(r_1; r_2)\}$.
• We take countably many homeomorphic copies $T\times\{n\}$ of the space $T$. We add two new points $a$, $b$ with local neighborhood bases $\{\bigcup T_m; m\ge n\}\cup \{a\}\subseteq B$ and $\{\bigcup T_m; m\ge n\}\cup \{b\}\subseteq B$.
• In this space we identify $(x,r_2,n)$ and $(x,r_2,n+1)$ for any $x\in R_1\setminus\{r_1\}$ and any even $n$. We also identify $(r_1,x,n)$ and $(r_1,x,n+1)$ for every odd $n$ and every $x\in R_2\setminus\{r_2\}$.

Let us call the resulting space $Q$.

Now for any space $Z$ we define a space $Q(Z)$ on the set $Z\times Q$ where a subset $B$ of $Z\times Q$ is open in $Z\times Q$ if and only if the following holds:

• If $(z;x)$ is an element of $B$, then there is a neighborhood $U$ of $x$ in $Q$ with $\{z\}\times U\subset B$.
• If $(z;a)$ is an element of $B$, then there is a neighborhood $U$ of $z$ in $Z$ with $U\times\{a\}\subset B$.

If we identify in the above space $Z\times Q$ all points of the set $Z\times\{b\}$, then we obtain a space $Q(Z)$.

Definition of $Q(Z)$. The space $Q(Z)$ contains a homeomorphic copy of $Z$. If $f$ is a continuous map from $Q(Z)$ to $Y$, then $f$ is constant on $Z$.

Definition of $X$. Let $X_0$ be a singleton. By induction we define $X_{n+1}=Q(X_n)$. Then $X_0\subset X_1\subset X_2\subset \dots $ are regular spaces. Let a subset of $X=\bigcup\{X_n; n=0,1,\dots\}$ be open if and only if $B\cap X_n$ is open for every $n$. The space $X$ is a regular space. Every continuous map from $X$ to $Y$ is constant.

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.

EDIT: I have put my attempt to translate the article here (let me know if you find any typos or mistranslations).

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.