What is the basic idea of Topology

Question

I am currently studying mathematics at a master's level, but oddly my first topology course ever will be next term... So I'm wondering some basic things:

$1)$ What role does topology serve in an overall view of mathematics (for example, measure theory can be seen as a certain abstraction of real analysis to measure spaces)

$2)$ What do people mean when they say "The topology on"? For example "The topology on $\mathbb{R}$", or the topology induced by a certain metric, etc

$3)$ What would be a good source of pre-requisite self study?

Answer 1

1. I recommend looking into the various implications Tychonoff's theorem has; on the most basic level I'm aware of - this theorem is equivalent to the axiom of choice within our working system. This theorem simply states the a product of compact space is compact, equipped with the correct topology.

2. A topology $\tau$ on a set $X$ is a collection of subsets of $X$ with the following properties;

a. $\emptyset , X \in \tau$

b. the union of elements in $\tau$ is again an element in $\tau$

c. the finite intersection of elements is again an element.

I recommend thinking about these axioms in context of the familiar open ball toplogy on $\mathbb{R}$.

3. Munkres Topology is extremely accessible and covers a nice chunk of point set topology