Metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold. Mathematics department stanford university math 61cm metric spaces we have talked about the notion of convergence in r. Properties of open subsets and a bit of set theory16 3. Each compact metric space is complete, but the converse is false. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. Introduction to metric and topological spaces oxford. Without taking a position for or against the current reforms in mathematics teaching, i. You have met or you will meet the concept of a normed vector space both in algebra and analysis courses. A point z is a limit point for a set a if every open set u containing z. Mathematics 490 introduction to topology winter 2007 1.
The outer space, denoted x n or cv n, comes equipped with a natural action of the group of outer automorphisms. Diameter of metric spaces mathematics stack exchange. A metric space consists of a set x together with a function d. He introduces open sets and topological spaces in a similar fashion. X r, we say that the pair m x, d is a metric space if and only if d satisfies the following.
I am trying to find a definition for the open cover of a metric space, but i cannot find it. These notes are collected, composed and corrected by atiq ur rehman, phd. A metric space is called complete if every cauchy sequence converges to a limit. A particular case of the previous result, the case r 0, is that in. Metricandtopologicalspaces department of pure mathematics. Definition and fundamental properties of a metric space. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. In this video, we have discussed the metric space with examples on metric space namaste to all friends, this video lecture series presented by vedam institute of mathematics is.
Probabilistic metric space encyclopedia of mathematics. Metric space more examples on metric space in hindilecture. For all of the lecture notes, including a table of contents, download the following file pdf. Mod05 lec23 limit and continuity of a function defined on a metric space duration. The lecture notes were taken by a student in the class. In the mathematical subject of geometric group theory, the cullervogtmann outer space or just outer space of a free group f n is a topological space consisting of the socalled marked metric graph structures of volume 1 on f n. Metric space more examples on metric space in hindi.
A metric space is just a set x equipped with a function d of two variables. Muhammad ashfaq ex hod, department of mathematics, government college sargodha. The first goal of this course is then to define metric spaces and continuous. We also establish some fixed point theorems for selfmappings defined on such spaces. Mod05 lec23 limit and continuity of a function defined on a metric space. Theorem in a any metric space arbitrary intersections and finite unions of closed sets are closed.
A metric induces a topology on a set, but not all topologies can be generated by a metric. Mar 05, 2018 in this video, we have discussed the metric space with examples on metric space namaste to all friends, this video lecture series presented by vedam institute of mathematics is useful to all. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. A metric space m m m is called complete if every cauchy sequence in m m m converges. A metric space is a set xtogether with a metric don it, and we will use the notation x. Real analysismetric spaces wikibooks, open books for an. Knowing whether or not a metric space is complete is very useful, and many common metric spaces are complete. Most fields of mathematics deal with sets carrying some addi. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. Turns out, these three definitions are essentially equivalent. The triangle inequality is geometrically obvious, but requires an analytical proof see section 7. A point p is a limit point of the set e if every neighbourhood of p contains a point q. In mathematics, a space is a set sometimes called a universe with some added structure while modern mathematics uses many types of spaces, such as euclidean spaces, linear spaces, topological spaces, hilbert spaces, or probability spaces, it does not define the notion of space itself.
A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. Math 3402 metric space topology school of mathematics and. Here d is the metric on x, that is, dx, y is regarded as the distance from x to y. For instance, r \mathbbr r is complete under the standard absolute value metric, although this is not so easy to prove. Sutherland often uses a lengthy series of examples of increasing difficulty to illustrate abstract concepts. The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The following properties of a metric space are equivalent. These are actually based on the lectures delivered by prof. Metric space, in which the distances between points are specified by probability distributions cf. A subset s of the set x is open in the metric space x.
In mathematics, a metric space is a set together with a metric on the set. On few occasions, i have also shown that if we want to extend the result from metric spaces to topological spaces, what kind. Although strictly speaking the metric space is the pair x, d it is a common practice to refer to x itself as being the metric space, if the metric d is understood from context. In his discussion of metric spaces, we begin with euclidian n space metrics, and move on to discrete metric spaces, function spaces, and even hilbert sequence spaces. Lecture notes analysis ii mathematics mit opencourseware. Suppose x, d is a metric space with the metric topology. For all of the lecture notes, including a table of contents, download the following file pdf 1. Abstract while modern mathematics use many types of spaces, such as euclidean spaces, linear spaces, topological spaces, hilbert spaces, or probability spaces, it does not define the notion of space itself. Metric spaces in pure and applied mathematics uni bielefeld. Often, if the metric dis clear from context, we will simply denote the metric space x.
Lecture notes assignments download course materials. Mathematics department stanford university math 61cm metric. So, if x is a metric space and a is a subset of x, then what is the definition for open cover of a. In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. U nofthem, the cartesian product of u with itself n times.