Math and science::Topology
Compactness. Motivation
What would we have to assume about a topological space
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 , the domain can be covered by a [...], each of which has an image that is bounded. - This is restricted to a given
. Can we generalize? - Every continuous
induces a neighbourhood around each , and will be bounded for each of these neigbourhoods. (Remember, the codomain is ). - 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
will induce an arbitrary open cover on (by [definition of what?]), and we impose that this cover has a finite subcover.
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