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Transcript
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.
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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 .
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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.
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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.
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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 .
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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.
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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.
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