How to understand compactness? How to understand the compactness in topology space in intuitive way?
Maybe you should think about compactness, as something that takes local properties to global properties. For example, if $f:K\rightarrow \mathbb{R}$ is continuous, $K$ is compact, and $f(x)&gt;t_x&gt;0$ for all x, then you can find $t&gt;0$ such that $f(x)&gt;t&gt;0$ for all x - so from $f(x)&gt;t_x&gt;0$ point wise, you know that $f&gt;t&gt;0$ as a function. (This is a simple consequence of Weierstrass theorem in $[a,b]\rightarrow \mathbb{R}$)

Usually we find some property that is true for every "small" enough open sets, then use compactness to reduce the case to finitely many open sets and use induction to show that the property is true for all of the space. This is at least how I understand compactness.



As the commenters below this message wrote (and I didn't emphasize enough), we usually use compactness to reduce infinite problems\conditions\restraints to a finite subset that cover the entire space, and then use some argument that works only for finite cases (like induction, taking max\min, take finite sums etc). In my example above, we wanted to find a minimum over all the lower bounds, but in the infinite case this is usually just an infimum (and can be zero), but when we reduce to a finite case there is a minimum.

In this way we can think of compactness as something that let us use some finite argument on infinite covers (and many times to transfer some property from the cover to the entire space).
For me, the compactness of a topological space means that it has enough points to provide exact solutions to continuous equations. More precisely,


  compactness = Any equation that can be approximated by a consistent system of $\leq$ inequalities of continuous functions has a solution.


For instance, being a solution to the equation $x^2 = 2$ is equivalent to being a simultaneous solution to the infinite system of inequalities $\lbrace |x^2-2| \leq \frac 1 n \rbrace_{n\in\mathbb N}$. This system is consistent in the sense that every finite subsystem has solutions; after all, we can approximate $\sqrt 2$ by rational numbers arbitrarily well. Then, it is the compactness of bounded intervals of real numbers that tells us that there exists a simultaneous solution of all of these inequalities, and hence of the equation itself.

In contrast, trying to encircle $\sqrt 2$ as a rational number fails, you can enclose it by smaller and smaller intervals, but when you try to catch it with an infinite number of intervals, it "falls through the cracks".

The principle discussed above is also known as the finite intersection property. The inequalities, or "conditions", define a collection of closed sets and the exact solution we seek is contained in the intersection of infinitely many closed sets. If every finite intersection of these closed sets is non-empty, i.e. if every finite collection of conditions can be fulfilled, then one could expect that all of the conditions can be satisfied simultaneously. This is precisely what compactness ensures.

Other examples:


Peano's theorem about the existence of solutions to ordinary differential equations relies on compactness to construct a good candidate.
A commonly used technique for showing the existence of solution for partial differential equations is the calculus variations. In this context, the weak compactness of the unit ball in a Hilbert space is often used to get a weak solution to the PDE which is then upgraded to a strong solution.