Trying to learn topology and with this proof

[Ed Quanta]

If S is a set with the discrete topology and f:S->T is any transformation of S into a topologized set T, then f is continuous.

Can someone help me prove this? I have no idea where to even begin.

 

[matt grime]

Def: a map, f, is continuous iff the inverse image of every open set is open. Let U be any subsey of T, f^{-1}(U) is a subset of S. All subsets of S are...?

Just use the definition of continuous

 

[mathwonk]

intuitively, "f is continuous" means that if x is close to a then f(x) is close to f(a). In a discrete topology, no two different points are ever close together.

So the only requirement for continuity is that, if two points x,a are close, i.e. if they are equal, then the values f(x) and f(a) should be close. That is pretty easy.

 

[NateTG]

If S is a set with the discrete topology and f:S->T is any transformation of S into a topologized set T, then f is continuous.

Can someone help me prove this? I have no idea where to even begin.

Well, you should start with the definition of continuous.

If you can't figure things out from there, here's a hint: Are there any subsets of S that are not open sets in the discrete topology?