Topology(meaning) When we define Topology we say that a topology on a set(let's say X) is a collection of subsets of X having certain 3 properties. Now, here what do we actually mean by saying "topology on a set". What is the geometrical picture of this topology? 
 A: I wish there was a simple answer to your question. A simple enough answer might be worthy of a serious award. 
There are two problems. The first is that the intuition of topology involves learning hard theorems. Compactness (an even less intuitive concept) and Connectedness are critical to our intuition about topologies. Separation axioms help us categorize spaces by how friendly they are, but involve tough proofs.  Geometry comes later, almost as an application in the nice settings, like locally Euclidean spaces.
Problem two is that there are some really scary topological spaces. People talk about "closeness" or "nearby" as related to topology, and they're right. But what does that really mean in spaces that badly fail T1, or some other basic structure? It's hard to say. And even harder to visualize. There are tricks, but things like the sorgenfrey plane (https://en.m.wikipedia.org/wiki/Sorgenfrey_plane) are so wildly different than "nicer" structures it makes it generalization quite hard. 
Long story short, keep working and thinking about it, and don't be discouraged by lack of clarity at first. As I was said to me, I'll say to you "learn the fundamentals." There is no other way. 
