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Math and science::Topology

Compactness. Motivation

What would we have to assume about a topological space \( X \) in order to prove that every continuous map \( f : X \to \mathbb{R} \) is bounded?

Reasoning summary

Condensed version.

  • What is the definition of being bounded?
  • What are some cases where we know functions are bounded?
  • The most basic case: the domain is [...].
  • Can we generalize this?
  • A little more general:
    For a given \( f \), the domain can be covered by a [...], each of which has an image that is bounded.
  • This is restricted to a given \( f \). Can we generalize?
  • Every continuous \( f \) induces a neighbourhood around each \( x \in X \), and \( f \) will be bounded for each of these neigbourhoods. (Remember, the codomain is \( \mathbb{R} \)).
  • So, by the definition of continuity, we have a cover where each subset is bounded.
  • Sadly, this set of neighbourhoods could be [...].
  • Thus we arrive at our requirement: every open cover must have [...].
  • Which means, every continuous \( f \) will induce an arbitrary open cover on \( X \) (by [definition of what?]), and we impose that this cover has a finite subcover.