$Y$ is $T_1$ iff there is regular space $X$ s.t. all continuous function from $X$ to $Y$ is constant

Question

I am trying to understand this article. This article states that $Y$ is $T_1$ space iff there exists a regular space $X$ (having at least two points), such that every continuous map from $X$ to $Y$ is constant.

I do not understand the construction of space $X$. Please anyone help me to show what regular space $X$ that satisfies that condition if $Y=\mathbb{R}$.

Answer 1

Eric van Douwen’s paper ‘A regular space on which every continuous real-valued function is constant’, Nieuw Arch. Wisk. $30$ $(1972)$, $143$-$145$, actually gives a ‘machine’ for starting with a $T_3$ space having two points that cannot be separated by a continuous real-valued function and producing from it a $T_3$ space on which all continuous real-valued functions are constant. I no longer have a copy of the paper or easy access to it, so I’m working from memory, but the construction below is either the same as or very similar to Eric’s.

Let $Z$ be any $T_3$ space with points $p$ and $q$ that cannot be separated by a continuous real-valued function. Let $Y=Z\setminus\{p,q\}$, and let $\kappa=|Y|$. Give $\kappa$ the discrete topology, and let $X=\kappa\times Z$ with the resulting product topology. For notational convenience let let $X_\xi=\{\xi\}\times Z$, $p_\xi=\langle\xi,p\rangle$, and $q_\xi=\langle\xi,q\rangle$ for $\xi<\kappa$. Clearly $\kappa$ is infinite, so there is a bijection $\varphi:\kappa\to\kappa\times Y\subseteq X$. We may (and do) further assume that if $\varphi(\xi)=\langle\eta,y\rangle$, then $\eta\ne\xi$.

Define an equivalence relation $\sim$ on $X$ as follows.

• Of course $x\sim x$ for all $x\in X$.
• We identify all copies of $p$ to a single point: $p_\xi\sim p_\eta$ for all $\xi,\eta<\kappa$.
• For all $\xi<\kappa$, $q_\xi\sim\varphi(\xi)$; this identifies each point of $\kappa\times Y$ with a unique $q_\xi$.

Let $X'=X/\!\sim$; we hope to show that $X'$ is $T_3$, and that every real-valued continuous function on $X'$ is constant.

Think of $Z$ as a string with endpoints $p$ and $q$. The idea of $\sim$ is to glue all of the copies of $p$ together into one point $p^*$, and then to run a separate string from $p^*$ to each other point, gluing the $q$ end of that string to the point. (I’ve always thought of this construction as Eric’s Spaghetti Machine.) This ought to ensure that if $x$ and $y$ are any points of $X'$, and $f:X'\to\Bbb R$ is continuous, the string from $p^*$ to $x$ ensures that $f(x)=f(p^*)$, while the string from $p^*$ to $y$ ensures that $f(y)=f(p^*)$, so that $f(x)=f(y)$.

And in fact it does. Let $\pi:X\to X'$ be the canonical quotient map, and let $p^*=\pi(p_0)$ (which is of course the same as $\pi(p_\xi)$ for each $\xi<\kappa$). If $f:X'\to\Bbb R$ is continuous, then $f\circ\pi:X\to\Bbb R$ is continuous. Let $g=f\circ\pi$; then $g(p_\xi)=g(q_\xi)$ for all $\xi<\kappa$, since $g\upharpoonright X_\xi$ is a continuous real-valued function on $X_\xi$, which is homeomorphic to $Z$. Let $x'\in X'$ be arbitrary. Then $x'=\pi\big(\langle\eta,y\rangle\big)$ for some $\langle\eta,y\rangle\in\kappa\times Y$, and there is a $\xi\in\kappa\setminus\{\eta\}$ such that $\varphi(\xi)=\langle\eta,y\rangle$. Then $\pi(q_\xi)=\pi\big(\varphi(\xi)\big)=\pi\big(\langle\eta,y\rangle\big)$, so $f(x')=g\big(\langle\eta,y\rangle\big)=g(q_\xi)=g(p_\xi)=f(p^*)$, and $f$ is indeed constant on $X'$.

It’s actually harder to show that $X'$ is $T_3$.

Let $x'\in X'\setminus\{p^*\}$, and let $U$ be an open nbhd of $x'$. There is some $\langle\eta,y\rangle\in\kappa\times Y$ such that $x'=\pi\big(\langle\eta,y\rangle\big)$. Let $V_0=X_\eta\cap\pi^{-1}[U]$; $V_0$ is an open nbhd of $y$ in $X$. Let

$$K_0=\{\xi<\kappa:\varphi(\xi)\in V_0\}=\{\xi<\kappa:\pi(q_\xi)\in\pi[V_0]\}\;,$$

and let $$V_1=\bigcup_{\xi\in K_0}\big(X_\xi\cap\pi^{-1}[U]\big)\;.$$

In general, given $V_n$ for some $n\in\omega$, let

$$K_n=\{\xi<\kappa:\varphi(\xi)\in V_n\}=\{\xi<\kappa:\pi(q_\xi)\in\pi[V_n]\}\;,$$

and let $$V_{n+1}=\bigcup_{\xi\in K_n}\big(X_\xi\cap\pi^{-1}[U]\big)\;.$$

Finally, let $V=\bigcup_{n\in\omega}V_n$; then $V=\pi^{-1}[U]$.

There is an open nbhd $W_0$ of $\langle\eta,y\rangle$ in $X$ such that $\operatorname{cl}_XW_0\subseteq V_0$. For each $\xi\in\bigcup_{n\in\omega}K_n$ there is an open nbhd $G_\xi$ of $q_\xi$ in $X$ such that $G_\xi\subseteq X_\xi$, and $\operatorname{cl}_XG_\xi\subseteq\pi^{-1}[U]$. Given $W_n$ for some $n\in\omega$, let $L_n=\{\xi<\kappa:\pi(q_\xi)\in\pi[W_n]\}$, and let

$$W_{n+1}=W_n\cup\bigcup_{\xi\in L_n}G_\xi\;.$$

Then $W=\bigcup_{n\in\omega}W_n$ is open in $X$, and $W=\pi^{-1}\big[\pi[W]\big]$, so $\pi[W]$ is an open nbhd of $x'$ in $X'$. Let $H_0=\operatorname{cl}_XW_0$. Given $H_n$ for some $n\in\omega$, let $M_n=\{\xi<\kappa:q_\xi\in\pi[H_n]\}$, and let

$$H_{n+1}=H_n\cup\bigcup_{\xi\in M_n}\operatorname{cl}_XG_\xi\;.$$

Then $H=\bigcup_{n\in\omega}H_n$ is closed in $X$, and $\pi^{-1}\big[\pi[H]\big]=H$, so $\pi[H]$ is closed in $X'$. Clearly $\pi[W]\subseteq\pi[H]\subseteq U$, so

$$x'\in\pi[W]\subseteq\operatorname{cl}_{X'}\pi[W]\subseteq\pi[H]\subseteq U\;.$$

I leave it to you to modify the argument slightly to show that $X'$ is $T_3$ at the point $p^*$ as well.

Answer 2

Edit. My answer below are not precise:
Please read the comment posted by @BrianM.Scott after my answer to see what I really should have said,
as well as to a link to his detailed and self-contained answer to a related question (and for more references).

One such simple example is by Mysior,
A. Mysior, A regular space which is not completely regular
Proc. Amer. Math. Soc. 81 (1981), 652-653
http://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601748-4/
It is only a two-page paper with the example described on the first page, and a remark based on it on the second page showing the existence of a regular space such that every real-valued function defined on it is constant. Correction. The example by Mysior is of a regular space in which there are two points $a,b$ with the property that $f(a)=f(b)$ for every continuous real valued function $f$. This example could be used to construct a regular space on which every continuous real-valued function is constant. See the comment below by @BrianM.Scott. You may also see Exercises 2.7.17 and 2.7.18 in Engelking, General Topology (1989) (more references there).

Another one is the Tychonoff corkscrew, in Counterexamples in Topology, will try to find a link in a minute. This one used to be the standard example, but the example by Mysior is easier to follow.

Can't find the best link, but the Tychonoff corkscrew is based on the Tychonoff plank (it takes countably many Tychonoff planks and glues them together in a certain way). The Tychonoff plank is based on the product $[0,\omega]\times [0,\omega_1]$. These and some related examples (examples 86-91) are described in
Counterexamples in Topology (1970, 2nd ed. 1978) Lynn Steen and J. Arthur Seebach, Jr.