Question

Consider topological space (X,Tx) and (Y, Ty), where (Y,Ty) is Hausdorff, and continuous functions f:X→Y and g:X→Y. (a) Prove that if the composition g∘f:X→Y is continuous, then f:X →Y is continuous. (b) Prove that if the composition g∘f:X→Y is continuous, then g:X →Y is continuous. You can proceed to prove the statements (a) and (b) regarding the continuity of functions in this topological space.

Answer 1

To prove statement (a), assume that g•f:X→Y is continuous and show that f:X→Y is continuous. To prove statement (b), assume that g•f:X→Y is continuous and show that g:X→Y is continuous.

Step-by-step explanation:

To prove statement (a), we need to show that if the composition g•f:X→Y is continuous, then f:X→Y is continuous.

1. Assume that g•f:X→Y is continuous.
2. Let V be an open set in Y.
3. Since g•f is continuous, (g•f)^(-1)(V) is open in X.
4. Since f = (g•f)&inverse;, f is continuous.

To prove statement (b), we need to show that if the composition g•f:X→Y is continuous, then g:X→Y is continuous.

1. Assume that g•f:X→Y is continuous.
2. Let V be an open set in Y.
3. Since g•f is continuous, (g•f)^(-1)(V) is open in X.
4. Since (g•f)^(-1)(V) = f^(-1)(g^(-1)(V)), f^(-1)(g^(-1)(V)) is open in X.
5. Since f is continuous, g^(-1)(V) is open in X.
6. Therefore, g is continuous.