6 Metric spaces Flashcards

1
Q

Metric, metric space, distance

A

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.

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2
Q

Open ball

A

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}

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3
Q

Equivalent metrics

A

Metrics on a set X are said to be equivalent if they induce the same topology on X.

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4
Q

Hausdorff space (T_2 space)

A

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.

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5
Q

Metrizable space

A

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.

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6
Q

To converge to / convergent

A

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.

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7
Q

Cauchy sequence

A

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.

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8
Q

Completeness

A

A metric space (X, d) is said to be complete if every Cauchy sequence in (X, d) converges to a point in (X, d).

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9
Q

Subsequence

A

If {xn} is any sequence, then the sequence x_n1, x_n2, … is said to be a subsequence if n1 < n2 < n3 < …

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10
Q

Increasing sequence / decreasing sequence

A

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.

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11
Q

Monotonic sequence

A

A sequence that is either increasing or decreasing.

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12
Q

Peak point

A

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

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13
Q

Completely metrizable space

A

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.

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14
Q

Separable space

A

A topological space is said to be separable if it has a countable dense subset.

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15
Q

Polish space

A

A topological space (X,T) is said to be a Polish space if it is separable and completely metrizable,

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16
Q

Souslin space (Suslin space)

A

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.

17
Q

Analytic set

A

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).

18
Q

Isometric spaces, Isometry

A

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.

19
Q

Isometric embedding

A

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).

20
Q

Completion

A

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).

21
Q

Banach space

A

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.

22
Q

Fixed point

A

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.

23
Q

Contraction mapping

A

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.

24
Q

Interior, interior point

A

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.

25
Q

Exterior, exterior point

A

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.

26
Q

Boundary, boundary point

A

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.

27
Q

Nowhere dense

A

A subset A of a topological space (X, T) is said to be nowhere dense if the set A^- has empty interior.

28
Q

Baire space

A

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.

29
Q

First category (meager), second category

A

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).

30
Q

Convex set

A

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.