>True or false: Let . Suppose each continuous function is bounded. Then is finite.

TIFR GS 2019, Part B Problem 3

General topology

Medium

Topology, Munkres

Do you really need a hint? Try it first!

Use the fact that every convergent sequence is bounded. So can you take a convergent sequence as a subset of \(\Bbb Q^2\) and name it as \(X\)?

Consider \(X = \{(1,0), (1/2,0), (1/3,0), …, (0,0)\}\). What can we say about the behaviour of \(f(x)\) as \(x\to (0,0)\)?

As \(f\) is continuous so \(f(x) \to f(0,0)\). Hence \(f(X)\) is convergent sequence. Now you are almost there!!

So, \(f\) is bounded by Hint 1. Hence we satisfy all the hypothesis of the question that any function on \(X\) is bounded but \(X\) is infinite. ### Hence the statement is false.

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