Trying to learn topology and with this proof

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Discussion Overview

The discussion revolves around proving that a function f from a set S with the discrete topology to a topologized set T is continuous. Participants explore the definition of continuity in topology and its implications in the context of discrete topologies.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant states that if S is a set with the discrete topology and f:S->T is any transformation, then f is continuous, seeking help to prove this.
  • Another participant defines continuity in terms of the inverse image of open sets and notes that all subsets of S are open in the discrete topology.
  • A different participant provides an intuitive understanding of continuity, suggesting that in a discrete topology, points are not close together, which simplifies the continuity requirement.
  • A later reply reiterates the need to start with the definition of continuity and questions whether there are any subsets of S that are not open in the discrete topology.

Areas of Agreement / Disagreement

Participants generally agree on the definition of continuity and the properties of discrete topologies, but the discussion remains unresolved regarding the formal proof of continuity.

Contextual Notes

Limitations include the assumption that participants are familiar with the definitions of topology and continuity, and the discussion does not resolve the specific steps needed for the proof.

Ed Quanta
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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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