## Metric Spaces

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dave470003  on June 6, 2012

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2nd Year University Of Warwick Mathematics Module - 2012

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# Metric Spaces

 MetricA metric d on a set M is a function d: MxM→R satisfying (for all x,y,z in M): 1. d(x,y) is positive 2. d(x,y) = 0 implies x = y 3. d(x,y) = d(y,x) 4. d(x,z) is less than or equal to d(x,y) + d(y,z)
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#### Definitions

Metric A metric d on a set M is a function d: MxM→R satisfying (for all x,y,z in M):
1. d(x,y) is positive
2. d(x,y) = 0 implies x = y
3. d(x,y) = d(y,x)
4. d(x,z) is less than or equal to d(x,y) + d(y,z)
Open Ball Centered at a, with radius r:
B(a,r) = {x in M: d(x,a)<r}
Closed Ball Centered at a, with radius r:
¬B(a,r) = {x in M: d(x,a)<r or d(x,a)=r}
Bounded A subset S of M is bounded if there exists an a in M and a positive r such that S is contained in B(a,r)
Norm A norm on a vector space V is a function ║·║:V→R such that for all x,y in V:
1. ║x║ is positive
2. ║x║ = 0 implies x = 0
3. ║cx║ = |c|║r║ for all c in R
4. ║x+y║ is less than or equal to ║x║+║y║
Convex A ball is called convex if:
║x║,║y║ are less than or equal to one implies that for every positive a and b which add to 1, ║ax + by║ is less than or equal to 1
Subspace (metric space) Let M be a metric space and H is a subset of M. (H, d_H) is a subspace of M where d_H(x,y) = d_M(x,y)
Open A subset U of M is open IN M if for all x in U, there exists a positive z such that B(x,z) is a subset of U.
Closed A subset U of M is closed IN M if its complement, M\U, is open (in M).
Convergence A sequence x_k in M is convergent to x in M if d(x_k,x) → 0.
Continuous Let (M_1,d_1), (M_2,d_2) be metric spaces and f:M_1→M_2. f is continuous at a if:
For every y > 0, there exists a z > 0 such that for every x in M_1, then d_1(x,a)<z implies d_2(f(x),f(a))<y.
f is continuous if it is continuous at every a in M_1.
Lipshitz Continuous Let (M_1,d_1), (M_2,d_2) be metric spaces and f:M_1→M_2. f is Lipshitz continuous if there exists a real c such that:
d_2(f(x),f(y)) is less than or equal to cd_1(x,y) for all x,y in M
Distance from a set Let A be a non-empty subset of a metric space M. Then the distance of x in M from A is:
d(x,A) := inf(d(x,z)) for all z in A.
Continuous (open sets relation for metric spaces) f:M_1→M_2 is continuous iff for every open subset U of M_2, ¬f(U) is open in M_1.
Topological equivalence Two metrics d_1, d_2 are called topologically equivalent if d_1 open and d_2 open sets collide.
Isometric The metric spaces (M_1, d_1) and (M_2, d_2) are called isometric if there exists a function f:M_1→M_2 such that:
1. f is bijective
2. d_2(f(x),f(y)) = d_1(x,y) for all x,y in M_1
Such a function is called an isometry.
Homeomorphic Two metric spaces M_1, M_2 are called homeomorphic if there exists an f:M_1→M_2 such that:
1. f is bijective
2. U is open in M_1 iff f(U) is open in M_2
Topological property A property P of metric spaces is a topological property (or topological invariant) if:
'M_1 satisfies P' and 'M_2 is homeomorphic to M_1' together imply that M_2 satisfies P.
TopologyA topology on a set T is a collection t of subsets of T such that:
1. The empty set and T are both in t.
2. The finite intersection of any sets in t is also in t.
3. The union of any sets in t is also in t.
(T,t) is called a topological space. Members of t are called open sets. A subset F of T is called closed if its complement in T (F\T) is open)
Basis A basis for a topology t on T is a collection of open sets (a subset of t) such that every member of t (every open set) is a union of members from B.
Sub-basis A sub-basis for a topology t on T is a collection of open sets (a subset of t) such that every member of t (every open set) is a union of finite intersections of members from B
Coarse topology Given t_1,t_2 as topologies on T, we say that t_1 is coarser than t_2 if t_1 is a subset of t_2.
Finer topology Given t_1,t_2 as topologies on T, we say that t_1 is finer than t_2 if t_2 is a subset of t_1.
Subspace (topology) Given (T,t) and S is a subset of T, define:
t_S = {(UnS): U belongs to t}
Then (S,t_S) is a subspace
Product space Let (T_1,t_1), (T_2,t_2) be topological spaces. Then the product topology t on T_1xT_2 has the basis:
B = {U_1 x U_2: U_1 belongs to t_1, U_2 belongs to t_2}
Neighbourhood A neighbourhood of x in T is a set H in T for which there exists an open set U which is a subset of H such that x is in U.
Closure Given H is a subset of T, its closure is given by:
¬H := {x belongs to T: every neighbourhood of x meets H}
(Informally, the least closed set containing H)
¬H = T\(T\H)°
Interior Given H is a subset of T, its interior is given by:
H° := {x belongs to T: H is a neighbourhood of x}
(Informally, the greatest open set within H)
H° = T\¬(T\H)
Boundry Given H is a subset of T, its interior is given by:
dH := {x belongs to T: every neighbourhood of x meets H and T\H}
dH = ¬H intersection ¬(T\H)
Continuous (open sets relation for topologies) f:T_1→T_2 is continuous iff for every open set U of T_2, ¬f(U) is open in T_1.
First projection Given the product space T_1xT_2, the first projection is given by:
pi_1: T_1xT_2 → T_1, pi_1(x,y) = x
Second projection Given the product space T_1xT_2, the second projection is given by:
pi_1: T_1xT_2 → T_2, pi_1(x,y) = y
More Complex Product For all j in J, let T_j be topological spaces. Then their product is:
1. The set T (which is the product of all T_j) of functions x on J such that x(j) belongs to T_j.
2. The coarsest topology on T for which the co-ordinate projections pi_j: T→T_j, pi_j(x) = x(j) are ALL continuous
Inductive (final) topology Given T_j (j belongs to J), a set S, and f_j: T_j→S, the final topology on S is the finest topology for which all f_j are continuous.
Quotient topology Given T, a set S, and a surjective function f: T→S, the finest topology for which f is continuous is called the quotient topology.
Metrizable (T,t) is called metrizable if there is a metric d on T such that t is exactly the collection of d-open sets.
Hausdorff property A topology has the Hausdorff property if for all disjoint x,y in T, there exist U and V - disjoint open sets such that x belongs to U and y belongs to V.
If a topology is metrizable it is Hausdorff.
Normal T is called normal if for all closed disjoint F_0, F_1 in T, there exist open disjoint G_0, G_1 such that F_0 is a subset of G_0 and F_1 is a subset of G_1
If a topology is metrizable it is normal.
F_sigma A subset S of T which is a union of countably many closed sets.
G_delta A subset S of T which is an intersection of countably many open sets.
F_sigma_delta A subset S of T which is an intersection of countably many F_sigma sets.
Dense S is called dense in T if ¬S = T
Nowhere Dense S is nowhere dense IN T if T\¬S is dense in T (i.e. ¬(T\¬S) = T
Meagre S, a subset of T, is called meagre if it is a union of a sequence of nowhere dense sets.
E.g:
Q is dense in R, and one point sets are nowhere dense, so Q is meagre.
Cover A cover of A is a collection U of sets whose union contains A. A cover is open if all its members are open.
Subcover A subcover of a cover U is a subcollection of u which still covers A.
Compact A topological space T is compact if EVERY open cover of T has a finite subcover.
Lower semi-continuous d:T→R is lower semi-continuous if for all c in R, {x in T: f(x)>c} is open.
Upper semi-continuous d:T→R is upper semi-continuous if for all c in R, {x in T: f(x)<c} is open.
Lebesgue number Given a cover U of a metric space M, a positive d is called a Lebesgue number of U if for all x in M, there exists a u in U such that B(x,d) is a subset of u.
Uniform continuity f: M→N (with M and N being metric spaces) is called uniformly continuous if:
For all positive e, there exists a positive d such that for all x,y in M, d_M(x,y)<d implies d_N(f(x),f(y))<e
Sequentially compact A metric space M is said to be sequentially compact if every sequence in M has a convergent subsequence (in M).
Relatively sequentially compact M, a subset of N, such that every x_j in M has a subsequence converging in N is called relatively sequentially compact.
Connected T is connected if the only decomposition of T into open (in T) sets is:
The empty set and T.
Decomposition A decomposition of a set T is two sets A,B such that A and B partition T.
Seperated T, a subset of S, is seperated by subsets U,V of S if:
1. T is a subset of U union V
2. U intersection V intersection T is the empty set (if S is a metric space, just U intersection V is the empty set)
3. U intersection T is nonempty
4. V intersection T is nonempty
Interval A set either of the form:
1. The empty set OR
2. {a} OR
3. [a,b] OR
4. [a,b) OR
5. (a,b] OR
6. (a,b)
Connected components Define x~y if x,y belong to a common connected subspace of T. This is an equivalence relation and the equivalence classes are called the connected components of T.
Path Let u,v belong to T. A path from u to v in T is a continuous function f: [0,1]→T such that f(0) = u and f(1) = v
Path connected T is called path-connected if any two points in T can be joined by a path in T. If T is path connected, T is connected.
Cauchy A sequence x_n in (M,d) is Cauchy if:
For every positive e, there exists a k in N such that for every i,j bigger than k, d(x_i,x_j) is less than e.
Complete (M,d) is complete if all Cauchy sequences converge in M.
Fixed point Let f: S→S. x (a point in S) is called a fixed point if f(x) = x.
Contraction A map f: (M,d)→(M,d) is a contraction if there exists a K in [0,1) such that for all x,y in M:
d(f(x),f(y)) is less than or equal to Kd(x,y)
Totally bounded (M,d) is totally bounded if for every positive e, there exists a finite F (a subset of M) such that every point of M has a distance less than e from any point of F
Equicontinuous Let C(M):= {f: M→M: f is continuous}. A subset S of C(M) is equicontinuous at x if for every positive e there exists a positive d such that |f(y) - f(x)| < e whenever f is in S and y is in B(x,d).
S is called equicontinuous if it is equicontinuous at every point in M.
Uniformly equicontinuous A subset S of C(M) is called uniformly equicontinuous if for every positive e there exists a positive d such that |f(y) - f(x)| < e whenever f is in S and d(y,x)<d, for all x in M.
Diameter Let S be a nonempty subset of (M,d). Then its diameter is given by:
diam(S) = sup(d(x,y)) for all x,y in S.
Completion (simple) A completion of (M,d) is a complete metric space N such that M is a dense subspace of N
Completion (modern) A completion of (M,d) is a complete metric space N together with an isometry i of M onto a subset of N such that i(M) is dense in N.
lp-norm An lp-norm is a norm function on an n-space satisfying:
║x║_p = (the sum of all (x_j)^p from j=1 to j=n)^(1/p)
Discrete metric Any set M with the metric:
d(x,y) = 0 if x=y
d(x,y) = 1 otherwise
Sunflower metric (french railways metric) The space R² with the metric:
d(x,y) = ║x-y║ if x and y lie on the same line passing through the origin
d(x,y) = ║x║+║y║ otherwise
Jungle river metric The space R² with the metric:
d(x,y) = |y_1 - y_2| if x_1=x_2
d(x,y) = |y_1| + |x_1 - x_2| + |y_2| otherwise
Properties of open sets 1. The intersection of any finite number of open sets in M is open in M.
2. The union of any number of open sets in M is open in M.
Theorem 1.9 (Relation between closedness and convergence of sequences) A subset F of a metric space M is closed iff for
every sequence x_k in F that converges to some x in M we
necessarily have that x in F .
Discrete topology A topology such that every subset of T (every set) is open.
Indiscrete topology A topology such that only the empty set and T are open.
Zariski topology A topology such that all closed sets consist of finite subsets of T, together with T.
Topology of pointwise convergence A topology on the set F(X) of real functions on a set X is deﬁned as the topology with a sub-basis formed by the sets
{f in F(X) : a < f(x) < b} (x in X, a, b in R).
Cantor setStep 0. Start with the interval [0; 1] and call it C0.
Step 1. Remove the middle third; two closed intervals remain.
. . .
Step N. From each of the 2^(N-1) remaining intervals remove the (open) middle third; 2^(N) closed intervals remain.
The set C (the intersection of all C_j for j=1 to j=infinity) is the (ternary) Cantor set.
It is a closed set with empty interior, so dC = C.
Notice that C has no isolated points.
It has uncountably many points, so many more than just the
end-points of the removed intervals.
Tietze's Theorem Every real-valued function deﬁned and continuous on a closed subset of a normal topological space T may be extended to a continuous function on the whole T .
Urysohn's Lemma Suppose that F_0, F_1 are disjoint closed subsets of a normal topological space T . Then there is a continuous function f : M→R such that:
f(x) = 0 for every x in F_0,
f(x) = 1 for every x in F_1.
If T is metrizable, such f may be deﬁned by the formula:
f(x) =
d(x, F_0)/d(x, F_0) + d(x, F_1)
Heine-Borel theorem Any closed bounded interval [a, b] in R is compact.
Fact 3.2 Any closed subset C of a compact space T is compact
Fact 3.3 Any compact subspace C of a Hausdorff space T is
closed in T .
Fact 3.4 A compact subspace C of a metric space M is bounded.
Corollary 3.6 Let all F_n be closed nonempty sets, and let F_n+1 be a subset of F_n. The intersection of all F_i from i=1 to i=infinity is nonempty.
Theorem 3.7 A continuous image of a compact space is
compact.
Theorem 3.8 A continuous bijection of a compact space T onto
a Hausdorff space S is a homeomorphism.
Tychonov's Theorem The product of compact spaces is compact.
Proposition 3.14 Every open cover U of a compact metric
space has a Lebesgue number
Theorem 3.15 A continuous map of a compact metric space M
to a metric space N is uniformly continuous.
Theorem 3.16 A metric space is compact iff it is sequentially
compact.

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