Motivation for topology

This is the most important thing on this Wiki probably, the motivation for topology.

Recall the definition of continuity, on a metric space

$$\forall a\in X\forall\epsilon>0\exists\delta>0:x\in B_\delta(a)\implies f(x)\in B_\epsilon(f(a))$$ . It seems natural to ask "what do we really need" from this definition. The open balls are open sets, any open set is the union of open balls. Can we go further?

So let us propose this:

$$\forall\text{open sets}\in Y,\ f^{-1}(\text{that open set})$$ is open. This looks very different from the definition.

How did we get to this? Well notice the

$$\forall a\in X$$ in the continuity definition, this means we're looking at every point, a lot of balls will overlap, and very soon you're not considering just open balls. The union of these balls is an open set though. (this motivates the "union of open sets is open" part of topologies)

Phrasing

My tutor told me "and very soon we realise we can discard this notion of distance" - I think this is too strong. We are not actively looking to discard distance, we find we can side-step it. We can get by without it.

This leads to the very important theorem (so important I have given it it's own page) Continuity definitions are equivalent