Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. The family Cof subsets of (X,d)defined in Definition 9.10 above satisfies the following four
A metric on a set M is a function d : M M ! R such that for all x; y; z 2 M, d(x; y) 0; and d(x; y) = The pair (M; d) is called a metric space. M and, if necessary, denote the distance by, for example, dM. The open ball centred at a 2 M with radius r is the set. the closed ball centred at a 2 M with radius r is fx 2 M : d(x; a) rg.
What is the difference between topological and metric spaces?
A metric space gives rise to a topological space on the same set (generated by the open balls in the metric). Different metrics can give the same topology. A topology that arises in this way is a metrizable topology.
A topological space is a set X with a collection of open subsets (including ∅ and X) that satisfying the two statements of the theorem. Many of our proofs for metric spaces would also apply more
A metric space (X;d) is a space Xwith a distance function d: X X!R+ (also called metric, from which the name metric space), that is a function which assigns to each pair of points x;y2Xa real number d(x;y) (their distance) and has the following properties:
of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. The first goal of this course is then to define metric spaces and continuous functions between metric spaces.
A metric space X with the collection open sets in the sense of a metric space is called the metric space topology on X. We note a few properties concerning the relationship between the open balls and open sets in a metric space X. First note that the metric space topology (X;T) is given by T = fO X: 8x 2 O; 9 open ball B s.t. x 2 B Og
Definition Given a metric space (X, d), a set E ⊆ X is bounded if . (i) A point p ∈ X is a limit point of the set E if for every r > 0, . that . (iii) E is open if . (iv) E is closed if . (Notice these definitions really only depend on knowing the neighborhoods.
We review basics concerning metric spaces from a modern viewpoint, and prove the Baire category theorem, for both complete metric spaces and locally compact Hausdor [1] spaces. The general idea of metric space appeared in [Fr echet 1906], and metric-space structures on vector spaces, especially spaces of functions, was
Metric Spaces A metric space is a pair (M,d) where M is a set of points and d is a metric that satisfies the following Positive definiteness:d(x,y) ≥ 0. Additionally d(x,y) = 0 if and only if x = y Symmetry: d(x,y) = d(y,x) Triangle inequality: d(x,z) ≤ d(x,y)+d(y,z) 3/30