Let f, g: X →Y be continuous functions. Assume that Y is...

Let $f, g: X \rightarrow Y$ 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 $x \in D$. Prove that $f(x)=g(x)$ for all $x \in X$.

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Key Concepts

Continuous Functions

Continuous functions are mappings where the preimage of every open set is also open. This property ensures that the structure of the space is maintained under the mapping, which is crucial when transferring local properties to the entirety of the space.

Hausdorff Space

A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods. This condition guarantees that limits of sequences (or nets) are unique, a property that is essential when proving that functions that agree on a dense subset must agree everywhere.

Dense Subset

A dense subset of a topological space is one whose closure equals the entire space. This means that every point in the space is either in the subset or is a limit of points from the subset, allowing properties verified on the dense subset to be extended to the entire space.

Uniqueness of Extension via Density and Continuity

The principle of uniqueness of extension via density and continuity states that if two continuous functions agree on a dense subset of a space, they agree everywhere. This is because the values of continuous functions at any point are determined by their behavior on nearby points, and in a Hausdorff space, the uniqueness of limits ensures the equality extends to the whole set.

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