Trying to learn topology and with this proof

In summary: So, in summary, if S has the discrete topology and f:S->T is any transformation into a topologized set T, then f is continuous because the only requirement for continuity in a discrete topology is that if two points x,a are equal, then the values f(x) and f(a) should be close. To prove this, one can start with the definition of continuity and consider if there are any subsets of S that are not open sets in the discrete topology.
  • #1
Ed Quanta
297
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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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  • #2
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
 
  • #3
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.
 
  • #4
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?
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of space and the relationships between objects within that space. It is concerned with the concept of continuity, connectedness, and convergence.

2. How is topology related to other areas of mathematics?

Topology has applications in various fields, including geometry, physics, computer science, and engineering. It is closely related to other areas of mathematics such as algebra, analysis, and geometry.

3. Why is topology important?

Topology is important because it provides a framework for understanding and analyzing complex structures and spaces. It has applications in many scientific fields, and it also helps to develop critical thinking and problem-solving skills.

4. How can I learn topology?

Learning topology requires a solid foundation in mathematics, including concepts such as sets, functions, and basic algebra. It is also helpful to have a good understanding of geometry and analysis. There are many resources available, such as textbooks, online courses, and video lectures, that can help you learn topology.

5. What are some common challenges when learning topology?

Some common challenges when learning topology include understanding abstract concepts, visualizing complex structures and spaces, and applying topological reasoning to practical problems. It is important to approach topology with patience, persistence, and a willingness to think creatively.

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