Simple example of a regular topological space that is not first countable or not metrisable?

Question

1. Is there an example of a regular topological space that is not first countable?

2. Is there an example of a regular topological space that is not metrisable?

Answer 1

Define a topology on $\mathbb N=\{1,2,3,\dots\}$ by calling a set $S\subseteq\mathbb N$ open if
$$\text{either}\quad1\notin S\quad\text{or else}\quad\sum_{n\in\mathbb N\setminus S}\frac1n\lt\infty.$$ With this topology $\mathbb N$ is a regular Hausdorff space, also zero-dimensional and normal, but not first-countable and therefore not metrisable.

Another example (if you know about ordinals): the ordinal space $[0,\omega_1]$ with the order topology.

Yet another example (if you know about product spaces): take some nontrivial compact Hausdorff space like $[0,1]$ or $\{0,1\},$ and then take the Tychonoff product of uncountably many copies of that space.

Answer 2

Let

$$\tau=\{U\subseteq\Bbb R:0\notin U\text{ or }\Bbb R\setminus U\text{ is finite}\}\;;$$

then $\langle\Bbb R,\tau\rangle$ is a compact Hausdorff space, so it’s regular. (In fact it’s the one-point compactification of a discrete space of cardinality $|\Bbb R|=\mathfrak{c}$.) It is not first countable, since the point $0$ has no countable base.

To see this, let $\mathscr{U}$ be any countable family of open nbhds of $0$. Then $\Bbb R\setminus U$ is finite for each $U\in\mathscr{U}$, and $\mathscr{U}$ is countable, so $\bigcup_{U\in\mathscr{U}}(\Bbb R\setminus U)$ is a countable subset of $\Bbb R$. $\Bbb R$ is uncountable, so there is an $x\in\Bbb R\setminus\bigcup_{U\in\mathscr{U}}(\Bbb R\setminus U)$. Let $V=\Bbb R\setminus\{x\}$; then $V$ is an open nbhd of $0$, but $x\in U\setminus V$ for each $U\in\mathscr{U}$, so no member of $\mathscr{U}$ is a subset of $V$, and $\mathscr{U}$ is therefore not a local base at $0$.

Every metrizable space is first countable, so this space is not metrizable.

Note that any uncountable set $X$ can be substituted for $\Bbb R$; $0$ is then replaced by some fixed $p\in X$.