A sequence x n in x is called a cauchy sequence if for any. Prove that the convergence is uniform if and only if for any convergent sequence x n. Math 118b final practice solutions charles martin march 20, 2012 1. Alternatively, one could show that for each i 1nthe set s i fx 2rn jx i 0gis. Recall that any set m can be given the discrete metric d d given by d dx. The distance function, known as a metric, must satisfy a collection of axioms. Cantor sets on topological spaces can be obtained as minimal sets with respect to discrete dynamical systems d. First of all, this space is classically very well understood see, e. This is a counterexample which shows that c2 would not necessarily hold if the collection werent nite. In the compact metric space xa sequence of functions f nnot necessarily continuousconverge pointwise to a continuous function f. Discrete metric space is often used as extremely useful counterexamples to illustrate certain concepts.
U nofthem, the cartesian product of u with itself n times. Xthe number dx,y gives us the distance between them. A disconnection of a set a in a metric space x,d consists of two nonempty sets a 1, a 2 whose disjoint union is a and each is open relative to a. A set is said to be connected if it does not have any disconnections example. Math 431 real analysis i homework due october 8 question 1. Then define a metric again called the supremum metric by d. Informally, 3 and 4 say, respectively, that cis closed under. For example, each subset of r is a metric space using dx.
A metric space is given by a set x and a distance function d. It is also sometimes called a distance function or simply a distance often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used we already know a few examples of metric spaces. Let b0, 1 be the set of all bounded functions on the interval 0, 1. A metric space x is compact if every open cover of x has a. N such that dx m,x n jan 18, 2016 mod05 lec24 continuous functions on a metric space nptelhrd. A norm kkis a function on a real vector space xto 0. Metric spaces are generalizations of the real line, in. The other metrics above can be generalised to spaces of sequences also. For example, the set of numbers 0, 1 is not open when considered as a subset of the real line because any open interval about the point 0 contains points not in 0, 1.
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. Complete and incomplete metric spaces part 1 duration. A metric space is a set equipped with a distance function, which provides a measure of distance between any two points in the set. But as we will see in examples it is often possible to assign different metrics to the. Let us look at some other infinite dimensional spaces. The vector space x c0, 1 is infinite dimensional since the functions pn. X, uis called an open subset of xif, for every p2u, there is some 0 such that b p. Real analysismetric spaces wikibooks, open books for an. Computable structures and operations on the space of. This can be shown directly, by nding an appropriate 0 for each x2r. A metric space is a pair x, d, where x is a set and d is a metric on x. In this section we will be studying the concept of.
Theorem in a any metric space arbitrary intersections and finite unions of closed sets are closed. Alternatively, one could show that for each i 1nthe set s i. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. A norm is a function on a real vector space x to 0. A pathcomponent of x is an equivalence class of x under the equivalence relation which makes x equivalent to y if there is a path from x to y. Often, if the metric dis clear from context, we will simply denote the metric space x. A of open sets is called an open cover of x if every x. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.
Im not giving proofs because others here have already done that. A path from a point x to a point y in a topological space x is a continuous function. Each of the following is an example of a closed set. In general, in a metric space such as the real line, a continuous.
These notes are collected, composed and corrected by atiq ur rehman, phd. Definition and fundamental properties of a metric space. A metric space is a set xtogether with a metric don it, and we will use the notation x. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. However, if 0, 1 is considered to be the entire space x, then it. The distinguished point is sometimes denoted 0 \displaystyle 0 due to its similar behavior to zero in certain contexts. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. The following properties of a metric space are equivalent. However, the supremum norm does not give a norm on the space c a, b of continuous functions on a, b, for it may contain unbounded functions. A metric space is a mathematical object in which the distance between two points is. In mathematics, a metric space is a set together with a metric on the set.
If x 1,d 1 and x 2,d 2 are metric spaces, a function f. Note that a pointed metric space is just a nonempty metric space with attention drawn to its distinguished point, and that any nonempty metric space can be viewed as a pointed metric space. One represents a metric space s s s with metric d d d as the pair s, d s, d s, d. If we take each c k rational, then the sum telescopes and s b a. Recall that in the exercise we showed that there are many continuous functions in x. A point p is a limit point of the set e if every neighbourhood of p contains a point q. Turns out, these three definitions are essentially equivalent. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. A pathconnected space is a stronger notion of connectedness, requiring the structure of a path. These are actually based on the lectures delivered by prof.
Consider the rst quadrant of the plane with usual metric. Defn a subset c of a metric space x is called closed if its complement is open in x. Each interval open, closed, halfopen i in the real number system is a connected set. Jan 02, 2017 a video explaining the idea of compactness in r with an example of a compact set and a noncompact set in r. A function f is called continuous at x 2x if for every. Also, we give topology induced by this metric and some results obtained from. Then this is a metric on xcalled the discrete metric and we call x. Another important example is s c0,t the space of continuous func tions x. Metric spaces notes these are updated version of previous notes. Pdf the purpose of this paper is to introduce modular ametric spaces. Introduction when we consider properties of a reasonable function, probably the. Ais a family of sets in cindexed by some index set a,then a o c.