Topology

1) Let X and Y be topological spaces. A function f : X → Y is continuous if and only if f⁻¹ (C) is closed in X for every closed set C ⊂ Y.

2) Let f, g : X → Y be continuous functions. Assume that Y is Hausdorff and that there exists a dense subset D of X such that f(x) = g(x) for all x ∈D.

Prove that f(x) = g(x) for all x ∈ X.

- State the hypotheses
- State the conclusions
- Clearly and precisely prove the conclusions from the hypotheses
- Results presented earlier in the text may be used and must be clearly documented