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,