4 Compactness - Mathematics
) is a compact space, that is, K is compact as a subset in (K,T K) The following three results, whose proofs are immediate from the definition, give methods of constructing compact sets Proposition 4 1 A finite union of compact sets is compact Proposition 4 2 Suppose (X,T ) is a topological space and K ⊂ X is a compact set
Chapter 5 Compactness - Mathematics
Theorem 5 4 Each closed subset of a compact space is compact Proof: Let Abe a closed subset of the compact space Xand let O be an open cover of Aby open sets in X Since Ais closed, then XnAis open and O = O [fXnAg is an open cover of X Since Xis compact, it has a nite subcover, containing only nitely many members O 1;:::;O n of O and may
16 Compactness
compact sets are compact", and so on Proposition 3 2 Compactness is not hereditary Proof We already know this from previous examples For example (0;1) is a non-compact subset of the compact space [0;1] Also N is a non-compact subset of the compact space + 1 The previous exercise should lead you to think about de ning \hereditary
254 Compactness via Open Sets - MIT Mathematics
In ordinary parlance only “compact” is used ) Example 25 4A(c) above illustrates: the interval [0,4] is t-compact; and the collection C with the two added open intervals around 0 and 4 give an infinite open covering which has a finite subcovering Theorem 25 4 The Heine-Borel Theorem The closed box B = [−k,k]×[−k,k] in R2 is t
Compact Sets in Metric Spaces
Compact Sets in Metric Spaces Math 201A, Fall 2016 1 Sequentially compact sets De nition 1 A metric space is sequentially compact if every sequence has a convergent subsequence De nition 2 A metric space is complete if every Cauchy sequence con-verges De nition 3 Let >0 A set fx 2X: 2Igis an -net for a metric space Xif X= [ 2I B (x ): De
Chapter 10 Compactifications
Definition 2 1 A Hausdorff space is called if each point has a\B−\locally compact neighborhood base consisting of compact neighborhoods Þ Example 2 2 1) A discrete space is locally compact 2) is locally compact: at each point , the collection of closed balls centered at is a‘8 BB base of compact neighborhoods
Compact resolvents - Math User Home Pages
compact implies that the resolvent (T ) 1 is meromorphic, and is compact away from poles This is an example of perturbation theory The proof uses basic facts about compact operators The easier case of Ta symmetric operator on a Hilbert space is already useful In that case, T 1 is a normal compact operator, and the resolvent (T ) 1 is
5 Locally compact spaces
5 Locally compact spaces Definition A locally compact space is a Hausdorff topological space with the property (lc) Every point has a compact neighborhood One key feature of locally compact spaces is contained in the following; Lemma 5 1 Let Xbe a locally compact space, let Kbe a compact set in X, and let Dbe an open subset, with K⊂ D
Compact Spaces Connected Sets Open Covers and Compactness
Compact Spaces Connected Sets Relative Compactness Theorem Suppose (X;d) is a metric space and K Y X Then K is a compact subset of (X;d) if and only if K is a compact subset of (Y;d) So unlike with closed and open sets, a set is \compact relative a subset Y" if and only if it is compact relative to the whole space
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