6 Metric spaces Flashcards
Metric, metric space, distance
Let X be a non-empty set and d a real-valued function defined on X x X such that for a, b e X:
i) d(a,b) >= 0 and d(a,b) = 0 if and only if a = b;
ii) d(a,b) = d(b,a); and
iii) d(a,c) <= d(a,b) + d(b,c), [the triangle inequality] for all a,b and c in X.
Then d is said to be a metric on X, (X,d) is called a metric space and d(a,b) is referred to as the distance between a and b.
Open ball
Let (X, d) be a metric space and r any positive real number. Then the open ball about a e X of radius r is the B_r(a) = {x: x e X and d(a,x) < r}
Equivalent metrics
Metrics on a set X are said to be equivalent if they induce the same topology on X.
Hausdorff space (T_2 space)
A topological space is said to be a Hausdorff space if for each pair of distinct points a and b in X, there exist open sets U and V such that a e U, b e V, and U n B = /O.
Metrizable space
A space (X, T) is said to be metrizable if there exists a metric d on the set X with the property that T is the topology induced by d.
To converge to / convergent
Let (X, d) be a metric space and x1, …, xn, … a sequence of points in X. Then the sequence is said to converge to x e X if given any epsilon > 0 there exists an integer n0 such that for all n >= n0, d(x, xn) < epsilon. This is denoted by xn -> x. The sequence y1, y2, …, yn, … of points in (X,d) is said to be convergent if there exists a point y e X such that yn -> y.
Cauchy sequence
A sequence x1, x2, …, xn, .. of points in a metric space (X, d) is said to be a Cauchy sequence if given any real number epsilon > 0, there exists a positive integer n0, such that for all integers m >= n0 and n >= n0, d(xm, xn) < epsilon.
Completeness
A metric space (X, d) is said to be complete if every Cauchy sequence in (X, d) converges to a point in (X, d).
Subsequence
If {xn} is any sequence, then the sequence x_n1, x_n2, … is said to be a subsequence if n1 < n2 < n3 < …
Increasing sequence / decreasing sequence
Let {x_n} be a sequence in R. Then it is said to be an increasing sequence if x_n <= x_n+1, for all n e N. It is said to be decreasing sequence if x_n >= x_n+1 for all n e N.
Monotonic sequence
A sequence that is either increasing or decreasing.
Peak point
Let {x_n} be a sequence in R. Then n_0 e N is said to be a peak point if x_n <= x_n0, for every n>= n0
Completely metrizable space
A topological space (X, T) is said to be completely metrizable if there exists a metric d on X such that T is the topology on X determined by d and (X, d) is a complete metric space.
Separable space
A topological space is said to be separable if it has a countable dense subset.
Polish space
A topological space (X,T) is said to be a Polish space if it is separable and completely metrizable,
Souslin space (Suslin space)
A topological space (X, T) is said to be a Souslin space if it is a Hausdorff and a continous image of a Polish space.
Analytic set
If A is a subset of a topological space (Y, T1) such that with the induced topology T2, the space (A, T2) is a Souslin space, then A is said to be an an analytic set in (Y, T1).
Isometric spaces, Isometry
Let (X, d) and (Y, d1) be metric spaces. Then (X, d) is said to be isometric to (Y, d1) if there exists a surjective mapping f: X -> Y such that for all x1 and x2 in X, d(x1,x2) = d1(f(x1), f(x2)). Such a mapping f is said to be an isometry.
Isometric embedding
Let (X, d) and (Y,d1) be metric spaces and f a mapping of X into Y. Let Z = f(X), and d2 be the metric induced on Z by d1. If f: (X,d) -> (Z, d2) is an isometry, then f is said to be an isometric embedding of (X, d) in (Y, d1).
Completion
Let (X, d) and (Y, d1) be metric spaces and f a mapping of X into Y. If (Y, d1) is a complete metric space, f: (X,d) -> (Y, d1) is an isometric embedding and f(X) is a dense subset of Y in the associated topological space, then (Y, d1) is said to be a completion of (X,d).
Banach space
Let (N, || ||) be a normed vector space and d the associated metric on the set N. Then (N, || ||) is said to be a Banach space if (N, d) is a complete metric space.
Fixed point
Let f be a mapping of a set X into itself. Then a point x e X is said to be a fixed point of f if f(x) = x.
Contraction mapping
Let (X, d) be a metric space and f a mapping of X into itself. Then f is said to be a contraction mapping if there exists an r e (0,1) such that
d(f(x1),f(x2)) < r.d(x1, x2) for all x1 and x2 e X.
Interior, interior point
Let (X, T) be any topological space and A any subset of X. The largest open set contained in A is called the interior of A and is denoted by Int(A). Each point x e Int(A) is called an interior point of A.
Exterior, exterior point
Let (X, T) be any topological space and A any subset of X. The set Int(X\A), that is the interior of the complement of A, is denoted by Ext(A), and is called the exterior of A and each point in Ext(A) is called an exterior point of A.
Boundary, boundary point
Let (X, T) be any topological space and A any subset of X. The set A^- \ Int(A) is called the boundary of A. Each point in the boundary of A is called a boundary point of A.
Nowhere dense
A subset A of a topological space (X, T) is said to be nowhere dense if the set A^- has empty interior.
Baire space
A topological space (X, d) is said to be a Baire space if for every sequence {X_n} of open dense subsets of X, the set Intersection(X_n) is also dense in X.
First category (meager), second category
Let Y be a subset of a topological space (X, T). If Y is a union of a countable number of nowhere dense subsets of X, then Y is said to be a set of the first category or meager in (X, T). If Y is not first category, it is said to be a set of the second category in (X, T).
Convex set
Let S be a subset of a real vector space V. The set S is said to be convex if for each x,y e S and every real number 0 < L < 1, the point Lx + (1 - L)y is in S.
