Topologist's need for Compactness/Connectedness While learning about Compactness and Connectedness today in Topology, I was wondering where mathematicians would need to use such properties in research?
 A: Compactness is quite possibly the most important topological property, both in topology itself, and in other fields which build on topological results. The reason is that compactness allows us to attribute properties of finite sets to infinite sets. This comes from the definition: any open cover has a finite subcover. In very broad terms, a compact space must have behavior which is in some sense bounded or limited. Some examples of when this is useful are:


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*The extreme value theorem

*Various fixed point theorems

*Continuous forward images of compact sets are compact (this is a basic result which is important because not all properties are preserved by forward images).

*The Arzela-Ascoli Theorem

*Compactness implies that every sequence has a convergent subsequence, an important result, especially in analysis.

*Construction of measures


This is only a small sample of when compactness is useful. In my research, I recently obtained a result using the extreme value theorem. We wanted to show that a solution to a certain problem exists, so we proved that solving the problem was equivalent to the existence of a minimum of a continuous function over a compact space. The hard part was showing the space was compact, but one we did, the extreme value theorem guarantees that a minimizer exists, and hence there is a solution to my problem.
Connectedness is important because it forces all of the elements in a space to be related to each other in some way. Connectedness has important implications like the intermediate value theorem, but in my experience, connectedness is usually a property added on to other theorems in order to preclude pathological situations, or to ensure that the statement of the theorem makes sense (later in Munkres, you will learn about homotopy theory which just makes more sense in a path-connected space).
Both of these concepts are important to such a wide variety of mathematical areas that they are very well understood. I find it doubtful that any new research is being done directly on compactness or conectedness. But any new theorems which show that show sufficiency for one of these properties in various settings would be important. Furthermore, both of these properties are important considerations for many results in analysis, ODE's, differential topology, algebraic topology, functional analysis, and beyond.
