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Concept
Compact space
Summary
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.
One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.
The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space.
For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1).
Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering (the real number line), the sequence of points has no subsequence that converges to any real number.
Official source
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