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Let X be a topological space with the property that, for every topological space Y, every function f : X Y is continuous. Prove that X has the discrete topology. (Hint: Let Y be the space X but with the discrete topology.)

Let X be a topological space with the property that, for every topological space Y, every function f : X Y is continuous. Prove that X has the discrete topology. (Hint: Let Y be the space X but with the discrete topology.)

Advanced Engineering Mathematics
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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**Problem Statement** Let \( X \) be a topological space with the property that, for every topological space \( Y \), every function \( f: X \rightarrow Y \) is continuous. Prove that \( X \) has the discrete topology. (Hint: Let \( Y \) be the space \( X \) but with the discrete topology.) **Explanation** In this problem, you need to demonstrate that the topological space \( X \) must have a discrete topology if every function from \( X \) to any other topological space \( Y \) is continuous. The hint suggests considering the case where \( Y \) is the same set as \( X \) but equipped with the discrete topology, meaning every subset of \( Y \) is open. The goal is to show that this means \( X \) itself must already be discrete, i.e., every subset of \( X \) is open in the topology of \( X \).
Transcribed Image Text:**Problem Statement** Let \( X \) be a topological space with the property that, for every topological space \( Y \), every function \( f: X \rightarrow Y \) is continuous. Prove that \( X \) has the discrete topology. (Hint: Let \( Y \) be the space \( X \) but with the discrete topology.) **Explanation** In this problem, you need to demonstrate that the topological space \( X \) must have a discrete topology if every function from \( X \) to any other topological space \( Y \) is continuous. The hint suggests considering the case where \( Y \) is the same set as \( X \) but equipped with the discrete topology, meaning every subset of \( Y \) is open. The goal is to show that this means \( X \) itself must already be discrete, i.e., every subset of \( X \) is open in the topology of \( X \).
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