Students sometimes struggle with the Heine-Borel Theorem; the authors certainly did the first time it was presented to them. This theorem can be hard to. Weierstrass Theorem and Heine-Borel Covering Theorem. Both proofs are two of the most elegant in mathematics. Accumulation Po. Accumulation Points. Heine-Borel Theorem. October 7, Theorem 1. K C Rn is compact if and only if every open covering 1Uαl of K has a finite subcovering. 1Uα1,Uα2,,Uαs l.
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If Theorme is compact but not closed, then it has an accumulation point a not in S. In fact it holds much more generally than for subspaces of a cartesian space:. This gives the next version:. We need to show that it is closed and bounded. Sign up using Facebook. It turns out that this is all we need: Hence what remains is to show that S S is bounded.
According to Wikipediathe theorem was first proved by Pierre Cousin in I am confused as to what this theorem is actually saying. Sign up or log in Sign up using Google. Thus, T 0 is compact.
Heine-Borel theorem in nLab
Call this section T 1. Note that we say a set of real numbers is closed if every convergent sequence in that set has its limit in that set. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof. Kolmogorov spaceHausdorff spaceregular spacenormal space.
K-topologyDowker space. Since all the closed intervals are homeomorphic it is sufficient to show the statement for [ 01 ] [0,1]. Continuing in like heind yields a decreasing sequence of nested n -boxes:.
S S is compactS S is closed and bounded. Then S S is a compact topological space with the induced topology precisely if it is complete and totally bounded with the induced metric. In this terminlogy, what we need to show is that 1 1 is admissible. I have seen different statements of the Heine-Borel theorem, but here is one that encapsulates all of what it could possibly mean.
Cantor spaceMandelbrot space. Through bisection of each of the sides of T 0the box T 0 can be broken up into 2 n sub n -boxes, each of which has diameter equal to half the diameter of T 0. Hence, every accumulation point of S is in Sso S is closed. Then at least one of the 2 n sections of T 0 must require an infinite subcover of Cotherwise C itself would have a finite subcover, by uniting together the finite covers of the sections. Lemma closed interval is compact In classical gheorem We could also try to generalise Theorem to subspaces of other metric spaces, but this fails: This theorem refers thekrem to uniform properties of S Sand in fact a further generalistion is true:.
CW-complexes are paracompact Hausdorff spaces. It is thus possible to extract from any open cover C K of K a finite subcover. Kris 1, 8 Since closed subspaces of compact spaces are compact this implies that S S is compact.
Heine–Borel theorem – Wikipedia
S is closed and bounded S is compactthat is, every open cover of S has a finite subcover. Sign up using Email and Password. We do so by observing that the alternatives lead to contradictions:. The history of what today is called the Heine—Borel theorem starts in the 19th century, with the search for solid foundations of real analysis.
Since a is a limit point of SW must contain a point x in S. Assume, by way of contradiction, that T 0 is not compact.