**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.

It is given that X is a topological space. For every element p∈X there exists a continuous function…

Answered: Exercise 2.35. Suppose X is a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.

Expert Solution

Step 1

It is given that $X$ is a topological space. For every element $p \in X$ there exists a continuous function so that $f : X \to ℝ$ such that $f^{- 1} (0) = \{p\}$ . Prove that $X$ is Hausdorff.

Definition: Let $X$ be a topological space. The topological space $X$ is said to be Hausdorff, if given $p, q \in X$ and $p \neq q$ , there exists open sets $U$ and $V$ such that $p \in U$ and $q \in V$ and $U \cup V = \emptyset$ , that is, the sets are disjoint.