I'm just starting self-learning Topology, and I have a lot of doubts...
Consider a set $X$ (e.g. the real numbers), and an open subset $S \subseteq X$. Let $\tau = \{ X, \varnothing, S \}$ be a topology on $X$.
Reading this blog, it is stated that closeness of points of $S$ is assured by "[..]that infinite ability to get take narrower and narrower subsets around a point[..]".
But isn't closeness of points in $S$ just assured from the enunciation of $\tau$? Aren't we creating a different topology on $X$ by introducing those "narrower subsets"?
In my experience, if you have good visual/spatial ability, then thinking about distances and closeness can be very helpful in assimilating topological definitions, and in proving results. Pictures can help guide you.
Topology often involves the abstraction of concepts that seem kind of obvious in familiar spaces, like $\mathbb R, \mathbb R^2$. For example, to me, an open set $U$ in $\mathbb R$ is a set such that, no matter where I am in $U$, I can always move around a little bit while staying within $U$. In topology, this kind of idea gets abstracted into the fact that for any $x \in U \in \tau \subseteq 2^X$, there is an open set $B$ such that $x \in B \subseteq U$.
However, it's very important to realize that in topology, strictly speaking, there is no notion of distance or closeness. Thinking about distances can be helpful for intuition and guidance, as mentioned, but you can never, ever write anything down about distances, or closeness, because those concepts don't exist in topology.
You might think of topology like a game, where you try to abstract away concepts that are kind of obvious in familiar spaces, and define them without ever using the concept of distance. A lot of it amounts to just set theory.
Two other things you might find helpful to keep in mind:
A lot of other mathematical spaces---metric spaces, normed vector spaces, manifolds...---are topological spaces, so topology is hiding in the background in a lot of places. This is part of the reason why it matters.
The definition of a topology is so abstract and loose that it permits all kind of weird topological spaces that don't make a lot of sense. In fact, entire books have been written about weird counterexamples in topology. Depending on your goals, you might find it better not to worry too much about these weird spaces. Learn a couple, know that they're out there, but don't worry about them too much.