Download metric spaces

what everybody should know about
the topology of
metric spaces
Math245, September 2007
basics
(1) definition: for all x, y ∈ X
(i)
d(x, y) > 0 if x 6= y, and d(x, x) = 0
(ii)
d(x, y) = d(y, x)
(iii)
d(x, y) 6 d(x, z) + d(z, y)
Note: any subset of a metric space is a metric space.
Note: an open ball is a neighborhood of its center.
(2) definition: a set is open if it contains a neighborhood of its every point; a set is closed
if its complement is open.
(3) corollary: an open ball is open; a closed ball is closed.
(4) corollary: a finite union or intersection of open sets is open; a finite union or intersection
of closed sets is closed.
(5) corollary: any union of open sets is open.
(6) corollary: any intersection of closed sets is closed.
(7) definition: lim xn = x if lim d(xn , x) = 0. Note: this is equivalent to the usual –N
n→∞
n→∞
definition, which can be stated as follows. For any neighborhood of x there exists N
such that xn belongs to that neighborhood for all n > N .
(8) definition: a point is called a limit point of a set if it can be a limit of a sequence of
distinct points of this set.
(9) theorem: a set is closed iff it contains its every limit point.
(10) definition: the boundary of a set consists of all points that are limit points of this set
and limit points of its complement at the same time.
(11) corollary: a set is open iff it contains no boundary points; a set is closed iff it contains
all the boundary points.
(12) definition: a set is connected if it contains no proper (nonempty and not itself) subset
which is both open and closed.
completeness
(13) definition: a metric space is complete if every Cauchy sequence converges.
(14) corollary: a subset of a complete metric space is complete iff it is closed.
(15) The Contraction Mapping Theorem.
(16) idea: every metric space has a “unique” completion.
compactness
(17) a closed set is compact if every sequence contains a convergent subsequence.
(18) corollary: any finite union of compact sets is compact.
(19) corollary: any intersection of compact sets is compact.
(20) corollary: any decreasing intersection of compact sets is nonempty.
(21) corollary: any compact set is complete.
(22) theorem: a set is compact iff any open cover contains a finite subcover.
CONTINUITY
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continuous maps between metric spaces
definition: a map is continuous if it preserves limits (i.e. interchangeable with limits).
equivalent definition: the usual –δ definition.
equivalent definition: a map is continuous if the preimage of any open set is open.
equivalent definition: a map is continuous if the preimage of every closed set is closed.
corollary: an image of a compact set is compact.
warning: an image of an open set may not be open.
fact: a Lipschitz map is continuous.
definition (idea): a continuous map is called a homeomorphism if it is invertible and
the inverse is continuous. Two homeomorphic spaces are “equivalent” in topological
sense. Two metrics are called equivalent if they produce the same topology, i.e. the
same notions of open and closed sets.
the space of continuous functions (maps)
(31) If X and Y are metric spaces then C(X , Y) denotes the space of continuous functions
from X to Y. The space C(X , Y) is a metric space with the so called uniform or
supremum metric
d(f, g) = sup dY f (x), g(h)
x∈X
(32) fact: convergence in the sup-metric on C(X , Y) is the same as the uniform convergence.
(33) theorem (difficult and confusing): if X and Y are complete then C(X , Y) is complete.
(34) warning: usually C(X , Y) is not compact, and is not even “locally compact”. A space
is locally compact if every bounded closed set is compact.
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