- #1

- 861

- 43

## Main Question or Discussion Point

I was asking to myself what is the usefulness of a topology. I'd thought this question before and couldn't find results on the literature, perhaps I was not searching with the right terms.

So I started thinking that maybe a topology is a way of defining the domain, codomain and image of a function?

For this I imagined the following situation. Let the set be ##\mathbb{R}## and define a metric ##d## on it such that ##d(x,y) = 1## if ##x \neq y## and ##d(x,y) = 0## if ##x = y##. Such metric induces the topology ##\tau = \{ \emptyset, \mathbb{R} \}##, correct?

Does that mean only functions whose domain is either ##\mathbb{R}## or ##\emptyset## can be defined on such topological space?

Ok, probably I wouldn't have needed to go too far... but that scenario was what I thought in my attempt to understand this.

So I started thinking that maybe a topology is a way of defining the domain, codomain and image of a function?

For this I imagined the following situation. Let the set be ##\mathbb{R}## and define a metric ##d## on it such that ##d(x,y) = 1## if ##x \neq y## and ##d(x,y) = 0## if ##x = y##. Such metric induces the topology ##\tau = \{ \emptyset, \mathbb{R} \}##, correct?

Does that mean only functions whose domain is either ##\mathbb{R}## or ##\emptyset## can be defined on such topological space?

Ok, probably I wouldn't have needed to go too far... but that scenario was what I thought in my attempt to understand this.