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Continuous functions on LCH spaces Proposition: If X is a LCH space, C0 (X ) = Cc (X ) in the uniform metric. Lemma: If X is a LCH space and E ⊂ X , then E is closed if and only if E ∩ K is closed for every compact set K ⊂ X . Proposition: If X is a LCH space, C (X ) is a closed subset of RX in the topology of uniform convergence on compact sets. () Locally Compact Hausdorff Spaces March 1, 2010 1/7 σ-compactness X is σ-compact if X = ∪∞ i=1 Ki and each Ki is compact. Proposition: If X is a σ-compact, LCH space there is a sequence {Un }n of precompact open sets such that X = ∪n Un & Un ⊂ Un+1 . Moreover if f ∈ RX the sets ( 1 g ∈ RX : sup |g (x) − f (x)| < m x∈Un ) n, m ∈ N form a neighborhood base for f in the topology of uniform convergence on compact sets. This topology is 1st countable, and fn → f uniformly on compact sets if and only if fn → f on each Un . () Locally Compact Hausdorff Spaces March 1, 2010 2/7 Partitions of unity If X is a topological space and E ⊂ X a partition of unity on E is a collection {hα }α with hα ∈ C (X , [0, 1]) such that For each x ∈ X there is a neighborhood containing x where only finitely many hα ’s are non-zero. P α hα (x) = 1 for x ∈ E . A partition of unity {hα }α is subordinate to an open cover U of E if for each α there exists U ∈ U such that supp hα ⊂ U. () Locally Compact Hausdorff Spaces March 1, 2010 3/7 Proposition: Let X be a LCH space, K ⊂ X compact and {Uj }nj=1 an open cover of K . There is a partition of unity on K subordinate to {Uj }nj=1 consisting of compactly supported functions. () Locally Compact Hausdorff Spaces March 1, 2010 4/7 Let X be a topological space and F ⊂ C (X ). F is equicontinuous at x ∈ X if for every > 0 there is an open set U containing x such that |f (x) − f (y )| < for all y ∈ U and f ∈ F. F is equicontinuous if it is equicontinuous at each point x ∈ X . F is pointwise bounded if {f (x) : f ∈ F} is a bounded subset of R or C for each x ∈ X . () Locally Compact Hausdorff Spaces March 1, 2010 5/7 Arzelà-Ascoli Theorem I Let X be a compact Hausdorff space. If F is an equicontinuous, pointwise bounded subset on C (X ), then F is totally bounded in the uniform metric, and the closure of F in C (X ) is compact. () Locally Compact Hausdorff Spaces March 1, 2010 6/7 Arzelà-Ascoli Theorem II Let X be a σ-compact LCH space. If {fn } is an equicontinuous, pointwise bounded sequence in C (X ), there exists f ∈ C (X ) and a subsequence of {fn } that converges to f uniformly on compact sets. () Locally Compact Hausdorff Spaces March 1, 2010 7/7