Exercise 2.35. Suppose X is a topological space, and for every pe X there exists a continuous function f: X → R such that f-1 (0) = {p}. Show that X is Hausdorff.

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**Exercise 2.35**

Suppose \( X \) is a topological space, and for every \( p \in X \) there exists a continuous function \( f : X \to \mathbb{R} \) such that \( f^{-1}(0) = \{p\} \). Show that \( X \) is Hausdorff.
Transcribed Image Text:**Exercise 2.35** Suppose \( X \) is a topological space, and for every \( p \in X \) there exists a continuous function \( f : X \to \mathbb{R} \) such that \( f^{-1}(0) = \{p\} \). Show that \( X \) is Hausdorff.
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It is given that X is a topological space. For every element pX there exists a continuous function so that f:X such that f-10=p. Prove that X is Hausdorff.

Definition:  Let X be a topological space. The topological space X is said to be Hausdorff, if given p, qX and pq, there exists open sets U and V such that pU and qV and UV=, that is, the sets are disjoint.

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