Trying to learn topology and with this proof

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In a discrete topology, every subset is open, which simplifies the proof of continuity for any function f from a discrete set S to a topologized set T. The definition of continuity states that the inverse image of every open set in T must be open in S. Since all subsets of S are open, the inverse image f^{-1}(U) will always be open for any open set U in T. Therefore, any transformation f from S to T is continuous. This conclusion follows directly from the properties of the discrete topology.
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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.
 
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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
 
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.
 
Ed Quanta said:
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?
 

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