# Solve: Topological Space

Let X be a normed space with the topology induced by the norm. Show that || || : X ---> R is a continuous function on X.

Please show all of your work. Thank you.

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#### Solution Preview

A function f : X → Y from a topological space X to a topological space Y is continuous if for every open set V ⊆ Y the pre-image f^−1(V ) is open in X.

Consider the map f(x) = ∥x∥ from X to R. Let V be open in R. Then, for every y∈V there's a number r>0 such that the set B(y,r)={z∈R: |z − y| < r} is ...

#### Solution Summary

In this solution, a step by step response is provided in which a topological space is contextualized.

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