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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94 terms
Terms | 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. |
| Topology | A 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 set | Step 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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