Familiar spaces in which every one point set is $G_\delta$ but space is not first countable

Question

In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it?

Though there are answers posted on this site to the above question, I don't find the spaces familiar to what has been taught in the book up to Article-30

I am not able to find examples either. Is any help possible on familiar examples?

Answer 1

Another example (perhaps a bit more familiar with readers of Munkres's text) is $\mathbb R^\omega$ with the box topology.^*

• It is not first countable because you can "diagonalise" through any countable collection of open neighbourhoods of a point.

  Given $\mathbf x = ( x_n )_n $ and a collection $\{ U_i : i \in \mathbb N \}$ is open neighborhoods of $\mathbf x$, without loss of generality we may assume that $U_i = \prod_n ( a_n^{(i)} , b_n^{(i)} )$ where $a_n^{(i)} < x_n < b_n^{(i)}$. Taking $c_n = \frac{a_n^{(n)} + x_n }{2}$ and $d_n = \frac{x_n + b_n^{(n)}}{2}$ it follows that $V = \prod_n ( c_n , d_n )$ is an open neighbourhood of $\mathbf x$, but $U_n \not\subseteq V$ for each $n$.

• It is pretty easy to verify that points are G_δ. (Given $\mathbf x = ( x_n )_n$, set $U_i = \prod_n ( x_n - \frac{1}{i} , x_n + \frac{1}{i} )$ for each $i$, and note that $\bigcap_i U_i = \{ \mathbf x \}$.)

---

^*_{This space is first explicitly mentioned on p.117 of Munkres's text, and has a separate index entry. Its non-metrizability is shown on p.132 and its disconnectedness is shown on p.151, both before the stated exercise. It is also the subject of several exercises prior to Section 30. To someone going through the text, it should be "familiar".}

Answer 2

Hint: Glue uncountably many copies of $\mathbb R$ together at the origin

Answer 3

A very late answer: $\cal{D}(\Omega)$, the space of $C^\infty$ functions with compact support in an open $\Omega\subset \mathbb{R}^n$ with the distribution topology, seems to me (but what do I know) the most familiar space that isn't first countable. Proof that it's not first countable: Rudin Functional Analysis 6.9 tells us it's not metrizable; Rudin 1.9(b) tells us that, among topological vector spaces, a space is metrizable iff it has a countable local base.

Now, we are also told at 6.3 that sets of the type $\{\varphi\in{\cal D}(\Omega):|\varphi(x_m)|<c_m, m=1,2,3\ldots\}$ are open in $\cal{D}(\Omega)$. Enumerate a countable dense set in $\Omega$ as $\{x_m\}$, and let $$A_m = \{\varphi\in{\cal D}(\Omega):|\varphi(x_i)|<1/m{\rm\,\,for\,}i=1,2,\ldots,m\}$$ We should have $\{0\}=\bigcap A_m$ since any member of the latter intersection would be $C^\infty$ and $0$ on a dense subset of $\Omega$. Because ${\cal D}(\Omega)$ is a topological vector space, for any $\varphi\in{\cal D}(\Omega)$, we also have $\{\varphi\}=\bigcap (A_m+\varphi)$. Every point is thus a $G_\delta$.