How to picture a first countable space?

Question

I find myself forgetting what it means for a space to be first countable on a frequent basis.

This is unlike say other terminologies such as "Hausdorff space", where you can picture balls separating each points. When you define first countable, you have to first recall what a "local neighborhood" (or was it countable neighborhood? Open neighborhood?) is. And I find myself forgetting what that term exactly is as well.

What is a surefire way to remember the definition of first countable? What do you see when you picture a first countable space?

Answer 1

I picture a (countable) sequence of shrinking neighborhoods around a point in $\mathbb{R}^2$. (Specifically, the balls $B_{1/n} (x)$.) If I draw any blob around this point, I just have to wait a bit and my neighborhoods will shrink enough so that they are eventually contained in it.

If something is first countable, then you can check if it is second countable. (Being second countable is stronger.)

And for being second countable, the picture is that you have countable collection of open sets that get fine enough so that any open set can be described as a union of some of them. For this I again picture $\mathbb{R}^2$, and fuzzily the collection of $1/n$ balls around the rational points, and maybe imagine taking some wiggly open set and filling it out with this little balls.

Answer 2

Imagine a point and around it a sequence of ever smaller disks with center that point.

Whatever form you draw, if that point is in its interior there will be a disk small enough to be contained in that form, too.

If you need help to keep apart "first" and "second" countable, you could recall that the definition of 1st depends on what happens around 1 point.