WEBA metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Example 1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. More
Metric Spaces (Notes) - MathCity.org
WEBCONTENTS OR SUMMARY: Metric Spaces and examples. Pseudometric and example. Distance between sets. Theorem: Let (X,d) (X, d) be a metric space. Then for any x,y ∈ X x, y ∈ X, |d(x, A) − d(y, A)| ≤ d(x, y). | d (x, A) − d (y, A) | ≤ d (x, y). Diameter of a set. Bounded Set. Theorem: The union of two bounded set is bounded.
WEBDefinition 1.1 Metric Space Suppose that Xis a set, and d: X× X→ [0,∞) ¢ R. Then, dis a metric on Xif and only if the following properties hold for a,b,c∈ X: M1) Positivity. d(a,b) g 0; M2) Equality. d(a,b) = 0 ⇐⇒ a= b; M3) Symmetry. d(a,b) = d(b,a); M4) Triangularity. d(a,b) f d(a,c)+d(b,c) . The tuple (X,d) is a metric space.
8.1: Introduction to Metric Spaces - Mathematics LibreTexts
WEBIn this section we briefly consider functions defined on subsets of a metric space \((A,\rho)\) with values in a metric space \((B,\sigma)\). We indicate that \(f\) is such a function by writing \[ f:(A,\rho)\to(B,\sigma).
1.3: Metric spaces - Mathematics LibreTexts
WEBA metric space is a set with a metric on it. More formally, a metric space is a pair (X, d) where X is a set and d is a metric on X. The elements of X are called points of the metric space. Given two points A, B ∈ X, the value d(A, B) is called distance from A to B. Example 1.3.1 Discrete metric.
WEBWe will write (X,ρ) to denote the metric space X endowed with a metric ρ. If Y is a subset of X, then the metric space (Y,ρ| Y ×Y) is called a subspace of (X,ρ). Example 1. Let ρ(x,y) := |x − y| for x,y ∈ IR. Then (IR,ρ) is a metric space. The set IR equipped with this metric is called the real line. Example 2. Let IR2:= IR × IR ...
Metric Spaces: Subspaces and Product Spaces - Hobart and …
WEBAs an example, we can make the set of rational numbers, $\Q$, into a metric space $(\Q,d')$ using the subspace metric inherited from $(R,d)$. Let $A=\{x\in\Q\,|\,-\sqrt{2} < x < \sqrt{2}\}.$ Then $A$ is the intersection with $\Q$ of the open interval $(-\sqrt{2},\sqrt{2}\,)$, which is an open set in $(\R,d)$.
WEBLet us give some examples of metric spaces. Example 7.1.2: The set of real numbers R is a metric space with the metric. d(x;y) := jx. yj: Items (i)–(iii) of the definition are easy to verify. The triangle inequality (iv) follows immediately from the standard triangle inequality for real numbers: d(x;z) = jx. zj = jx.
Introduction to Metric Spaces | Mathematics - MIT OpenCourseWare
WEBThis course provides a basic introduction to metric spaces. It covers metrics, open and closed sets, continuous functions (in the topological sense), function spaces, completeness, and compactness.
Metric Spaces (Definition and Examples) | Introduction to Metric …
WEBThe constraint of p to Y × Y thus defines a metric on Y, which we refer to as a metric subspace. Hence, a metric space is a nonempty subset of Euclidean space, of an L P (E) space,1 ≤ p ≤ ∞, and of C[a, b].