Q:

A canonical example for spaces that aren't $1^{\text{st}}$ countable

To get the feeling for metric spaces (or $1^{\text{st}}$ countable spaces, in general), I always find that visualizing them as $\Bbb R^3$ gives a good intuition of what is going on (locally, of course).
However, a general topological space $\Bbb X$ need not be $1^{\text{st}}$ countable and I have hard time visualizing the behavior of the neighborhood base near any particular point $x\in\Bbb X$.

Is there any good canonical example of a topological space $\Bbb X$ that is NOT $1^{\text{st}}$ countable?
  How should I visualize the neighborhood of a point $x$ in such a space?

A:

Here are four examples that I find useful.
The first two are easy. Let $Y$ be an uncountable set and $p$ a point not in $Y$, and let $X=\{p\}\cup Y$. Points of $Y$ are isolated, and the open nbhds of $p$ are precisely the sets $X\setminus F$ such that $F$ is a finite subset of $Y$. In other words, the topology on $Y$ is discrete, and the topology at $p$ is the cofinite topology.
If you change the topology so that the open nbhds of $p$ are precisely the sets $X\setminus C$ such that $C$ is a countable subset of $Y$, you get the second example. If $|X|=\omega_1$, $p$ has a nested local base in the second example but not in the first.
A compact example in which every point has a nested local base is the ordinal space $$\omega_1+1=[0,\omega_1]$$ with the order topology; every point except $\omega_1$ has a countable local base, and the sets $(\alpha,\omega_1]$ for $\alpha<\omega_1$ form a nested local base at $\omega_1$. If you remove the set of countable limit ordinals from this space, you get the second example above.)
The fourth example starts with a free ultrafilter $p$ on $\Bbb N$. The space is $X=\{p\}\cup\Bbb N$. Points of $\Bbb N$ are isolated. A set $U\subseteq X$ is an open nbhd of $p$ if and only if $p\in U$ and $U\cap\Bbb N\in p$. For a much bigger example you can take $\beta\Bbb N$, the Čech-Stone compactification of $\Bbb N$; no point in $\beta\Bbb N\setminus\Bbb N$ has a countable local base. If $p\in\beta\Bbb N\setminus\Bbb N$, the subspace $\{p\}\cup\Bbb N$ of $\beta\Bbb N$ is precisely the space described at the beginning of this paragraph.

