Checking My Topological Result: Is f^{-1}(S') \subset T?

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In summary, the conversation discusses the idea that to show a function is continuous between topological spaces, it is sufficient to show that the inverse image of a basis for the codomain's topology is contained in the domain's topology. This is because every open set in the codomain's topology can be expressed as a union of sets in the basis, and the operations of inverse image and union commute. This idea is commonly used and is similar to the definition of continuity in calculus.
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quasar987
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I just discovered the following. But since half the things I find in topology turn out to be wrong, I feel I better check with you guys.

What I convinced myself of this time is that if you have a function f:(X,T)-->(Y,S) btw topological spaces, and S' is a basis for S, then to show f is continuous, is suffices to show that [itex]f^{-1}(S') \subset T[/itex].

In words, that is because every open set of S can be written as a union of sets of S', and the operations of f^{-1} and union commute. (and that a union of open sets is open)
 
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  • #2
That looks right. In fact, it's rather often used -- e.g. look at the definition of continuity you learned in calc I.
 
  • #3
it's a fact!
 
  • #4
Thanks.
 

Related to Checking My Topological Result: Is f^{-1}(S') \subset T?

1. What does it mean for f^{-1}(S') to be a subset of T?

When f^{-1}(S') is a subset of T, it means that all the elements in the preimage of S' (the set of all inputs that map to the elements in S') are also elements of T. In other words, every possible input that produces an output in S' is also a valid input for the function f in the set T.

2. How is the preimage of S' determined?

The preimage of S' is determined by finding all the inputs of the function f that map to the elements in the set S'. This can be done by solving the equation f(x) = y for each element y in S'.

3. What does it mean if f^{-1}(S') is not a subset of T?

If f^{-1}(S') is not a subset of T, it means that there are inputs that map to elements in S' but are not valid inputs for the function f in the set T. This could indicate an error in the topological result or that the set T is not complete enough to fully capture the behavior of the function f.

4. How do you check if f^{-1}(S') is a subset of T?

To check if f^{-1}(S') is a subset of T, you can compare the elements in the preimage of S' to the elements in the set T. If all the elements in the preimage are also in T, then f^{-1}(S') is a subset of T. You can also graph the function f and visually check if the preimage of S' falls within the set T.

5. Why is it important to check if f^{-1}(S') is a subset of T?

Checking if f^{-1}(S') is a subset of T is important because it verifies the accuracy of the topological result. It ensures that the set T accurately captures the behavior of the function f and that all possible inputs for the function are included. If f^{-1}(S') is not a subset of T, it could indicate an error in the topological result or the need for a more inclusive set T.

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