"A metric space is a set together with a notion of distance between its elements, usually called points."
A metric space is a set of points with a distance function satisfying certain properties. It is important in topology as it is a generalization of Euclidean space.
Sets and Set Operations: This topic covers the basic concepts of sets, set notation, and set operations.
Set Theory: This topic covers axioms of set theory, Commonly used set operations, Special sets (empty sets, finite sets, etc.), and transfinite induction.
Topological Spaces: This topic covers the definition of topological spaces, Metric Spaces, open sets, closed sets, continuity, compactness, connectedness, and separability in metric spaces.
Functions and Convergence: This topic covers the concept of continuity, limits, and convergence of a sequence of functions.
Basic Topology: This topic covers the notion of open and closed sets, compactness, connectedness, and closure.
Distance Functions and Metrics: This topic covers distance functions and metrics, metric spaces, and their properties. It also includes examples such as Euclidean space, discrete metric space, and more.
Completeness, Uniformity and Equivalence of Metrics: This topic covers the completeness of metric spaces, uniform continuity, and equivalence of metrics.
Sequences and Series in Metric Spaces: This topic covers sequences and series, limit points, convergence, and Cauchy sequences in metric spaces.
Topological Properties and Separation Axioms: This topic covers separation properties and axioms such as Hausdorff, regular, normal, and Urysohn's Lemma.
Topology of the Real Line: This topic covers real numbers, intervals, boundedness, and closure of intervals.
Compactness and Connectedness: This topic covers compact sets, connected sets, and their properties such as the Heine-Borel theorem.
Metric Space Constructions: This topic covers constructions of new spaces from existing ones using mappings such as quotient spaces, product spaces, and subspace topologies.
Metric Space Completeness and its Applications: This topic covers complete metric spaces and their applications such as the Baire Category theorem and the fixed-point theorem.
Metric Space Geometry: This topic covers properties of metric spaces such as length, angle, curvature, geodesics, and distance.
Metric Spaces in Analysis and Algebra: This topic covers the applications of metric spaces in analysis and algebra, including Banach spaces, Hilbert spaces, and normed spaces.
Complete metric spaces: A metric space is said to be complete if every Cauchy sequence converges to a limit within the space itself. Examples: Banach spaces, Hilbert spaces.
Incomplete metric spaces: A metric space is said to be incomplete if there exist Cauchy sequences that do not converge within the space. Example: The rational numbers under the usual metric.
Separable metric spaces: A metric space is said to be separable if it contains a countable dense subset. Example: The real line with the usual metric.
Non-separable metric spaces: A metric space is said to be non-separable if it does not contain a countable dense subset. Example: The space of continuous real-valued functions on a compact interval [a,b] under the supremum metric.
Compact metric spaces: A metric space is said to be compact if every open cover of the space has a finite subcover. Example: The closed unit ball in a finite-dimensional normed vector space.
Non-compact metric spaces: A metric space is said to be non-compact if it is not compact. Example: The real line with the usual metric.
Connected metric spaces: A metric space is said to be connected if it is not the disjoint union of two non-empty open sets. Example: The unit circle with the usual metric.
Disconnected metric spaces: A metric space is said to be disconnected if it is the disjoint union of two non-empty open sets. Example: The real line with the usual metric, where the two open sets are the sets of negative and positive real numbers.
Metric spaces with specific properties: There are many other types of metric spaces such as Banach spaces, Hilbert spaces, normed spaces, locally compact spaces, homogeneous spaces, etc., each with its own specific set of properties and applications.
"The distance is measured by a function called a metric or distance function."
"Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry."
"The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance." "Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane."
"A metric may correspond to a metaphorical, rather than physical, notion of distance." "For example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another."
"Since they are very general, metric spaces are a tool used in many different branches of mathematics."
"Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs."
"In abstract algebra, the p-adic numbers arise as elements of the completion of a metric structure on the rational numbers."
"Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces."
"Many of the basic notions of mathematical analysis, including balls, completeness, as well as uniform, Lipschitz, and Hölder continuity, can be defined in the setting of metric spaces."
"Yes, notions such as continuity, compactness, and open and closed sets can be defined for metric spaces."
"Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even more general setting of topological spaces."