Informally, 3 and 4 say, respectively, that cis closed under. Intro real analysis, lec 33, euclidean metric, triangle. The following properties of a metric space are equivalent. Xis compact, and x j2kis a cauchy sequence, then there exists x2ksuch that lim j. A metric space x is compact if every open cover of x has a finite subcover. Compact sets in metric spaces are complete mathonline. Note that every compact space is locally compact, since the whole space xsatis es the necessary condition. Let f n be a decreasing sequence of closed nonempty subsets of x, and let g n fc n. Compact sets in metric spaces uc davis mathematics. We will also be interested in the space of continuous rvalued functions cx. A of open sets is called an open cover of x if every x. Ais a family of sets in cindexed by some index set a,then a o c. Theorem each compact set k in a metric space is closed and bounded. Proof if k is a closed set in x, then k is compact lemma 5.
Say a metric space xis sequentially compact if every sequence in xhas a subsequence that converges in x. A pathconnected space is a stronger notion of connectedness, requiring the structure of a path. Pdf some notes on compact sets in soft metric spaces. So, lets look at an example of a subset of a metric space that is not compact. 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. Every continuous function on a compact set is uniformly continuous. R 0,1 from the real number line to the closed unit interval, and define a topology on k so that a sequence in k converges towards if and only if converges towards f x. A metric space is a set in which we can talk of the distance between any two of its elements. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. Intro real analysis, lec 33, euclidean metric, triangle inequality, metric spaces, compact sets. The latter result is due to pavel alexandrov and urysohn. A metric or topological space is compact if every open cover of the space has a nite subcover. A sequentially compact subset of a metric space is bounded and closed. Note that in metric spaces the notions of compactness and sequential compactness coincide.
Theorem in metric space, a subset kis compact if and only if it is sequentially compact. A set k in a metric space x, d is said to be compact if any open cover u. Proposition each closed subset of a compact set is also. Compact metric space yongheng zhang when we say a metric space xis compact, usually we mean any open covering of xhas a nite subcovering. In fact, every compact metric space is a continuous image of the cantor set.
Spaces of continuous functions if the underlying space x is compact, pointwise continuity and uniform continuity is the same. We then have the following fundamental theorem characterizing compact metric spaces. Every compact metric space is second countable, and is a continuous image of the cantor set. In other words, the compact sets in rn are characterized by the bolzanoweierstrass property. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. A metric space which is sequentially compact is totally bounded and complete. When we discuss probability theory of random processes, the underlying sample spaces and eld structures become quite complex.
A metric space is just a set x equipped with a function d of two variables which measures the distance between points. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Euclidean spaces are locally compact, but infinitedimensional banach. A metric space is compact if and only if it is sequentially compact. The definition below imposes certain natural conditions on the distance between the points. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. If you do not wish to use the heineborel theorem for metric spaces as suggested in the answer by igor rivin then here is another way of proving that a compact metric space is complete. A path from a point x to a point y in a topological space x is a continuous function.
In general, it may be more difficult to show that a subset of a metric space is compact than to show a subset of a metric space is not compact. Denition theinteriorof a, denoted inta, is the largest open set contained in a alternatively, the union of all open sets contained in a. Compact spaces connected sets open covers and compactness. A metric space which is totally bounded and complete is also sequentially compact. Denition theclosureof a, denoted a, is the smallest closed set containing a alternatively, the intersection of all closed sets containing a. In euclidean space, a set being compact is equivalent to. A metric space x is compact if every open cover of x has a. A video explaining the idea of compactness in r with an example of a compact set and a noncompact set in r. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. On the other hand, the closed unit ball of the dual of a normed space is compact for the weak topology. A metric space is sequentially compact if every sequence has a convergent subsequence.
A subset a of x is compact with respect to the subspace topology on a if. Suppose kis a subset of a metric space xand k is sequentially compact. Pdf in this article, we mainly formalize in mizar 2 the equivalence among a. Remarks in the theory of point set topology, the compactness implies the sequential compactness, but not vice versa. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. We do not develop their theory in detail, and we leave the veri. A subset, k, of m is said to be compact if and only if every open cover of k by open sets in m. A space is locally compact if it is locally compact at each point. So unlike with closed and open sets, a set is \compact relative a subset y if and only if it is compact relative to the whole space. Compact spaces connected sets relative compactness theorem suppose x. A metric space is said to be locally compact if every point has a compact neighborhood. Schep in this note we shall present a proof that in a metric space x. If uis an open cover of k, then there is a 0 such that for each x2kthere is a.
A metric space is complete if every cauchy sequence converges. X, there exists an open neighborhood u of x with closure u. Cauchy sequence in x has a convergent subsequence, so, by lemma 6 below. Turns out, these three definitions are essentially equivalent. The equivalence between closed and boundedness and compactness is valid in nite dimensional euclidean spaces and some special in nite dimensional space such as c1k.
700 1005 1558 1014 878 1253 1192 1248 777 203 1664 1487 1687 1256 422 896 466 841 460 833 975 1235 1526 42 1432 347 1559 894 694 565 990 900 618 926 970 812 340