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Appendix A Elements of topology and functional analysis Here, for the reader’s convenience, we collect fundamental concepts, definitions, and theorems used in this monograph, which are, however, rather standard and thus presented here mostly without proof, although some specific generalizations or modifications are accompanied by proofs. There are many textbooks and monographs on this subject, such as, e.g., [52, 84, 173, 175, 314, 344, 624]. A.1 Ordering A binary relation, denoted by Ä,onasetX is called an ordering if it is reflexive (i.e., x Ä x for all x 2 X), transitive (i.e., x1 Ä x2 & x2 Ä x3 imply x1 Ä x3 for all x1; x2; x3 2 X) and antisymmetric (i.e., x1 Ä x2 & x2 Ä x1 imply x1 D x2). The ordering Ä is called linear if x1 Ä x2 or x2 Ä x1 always holds for every x1; x2 2 X. An ordered set X is called directed if for every x1; x2 2 X, there is x3 2 X such that both x1 Ä x3 and x2 Ä x3. Instead of x1 Ä x2, we also write x2 x1.Byx1 < x2 we understand that x1 Ä x2 but x1 6D x2. Having two ordered sets X1 and X2 and a mapping f W X1 ! X2, we say that f is nondecreasing (respectively nonincreasing) if x1 Ä x2 implies f .x1/ Ä f .x2/ (respectively f .x1/ f .x2/). We say that x1 2 X is the greatest element of the ordered set X if x2 Ä x1 for every x2 2 X. Similarly, x1 2 X is the least element of X if x1 Ä x2 for every x2 2 X. We say that x1 2 X is maximal in the ordered set X if there is no x2 2 X such that x1 < x2. Note that the greatest element, if it exists, is always maximal but not conversely. Similarly, x1 2 X is minimal in X if there is no x2 2 X such that x1 > x2. The ordering Ä on X induces the ordering on a subset A of X, given just by the restriction of the relation Ä. We say that x1 2 X is an upper bound of A X if x2 Ä x1 for every x2 2 A. Analogously, x1 2 X is called a lower bound of A if x1 Ä x2 for every x2 2 A. If every two elements x1; x2 2 X possess both a least upper bound and a greatest lower bound, denoted respectively by sup.x1; x2/ and © Springer Science+Business Media New York 2015 579 A. Mielke, T. Roubícek,ˇ Rate-Independent Systems: Theory and Application, Applied Mathematical Sciences 193, DOI 10.1007/978-1-4939-2706-7 580 A Elements of topology and functional analysis inf.x1; x2/ and called the supremum and the infimum of fx1; x2g, then the ordered set .X; Ä/ is called a lattice. Then the supremum and the infimum exist for every finite subset and are determined uniquely because the ordering is antisymmetric. The so-called Kuratowski–Zorn lemma [343, 631] says that if every linearly ordered subset of X has an upper bound in X, then X has at least one maximal 1 element. Having a directed set and another set X, we say that .x /2 is a net in . / X if there is a mapping ! X W 7! x . Having another net xQ Q Q 2Q in X,we say that this net is finer than the net .x /2 if there is a mapping j W Q ! such Q Q that for every 2 ,wehavexQ Q D xj./Q , and moreover, for every 2 , there is Q 2 Q large enough that j.Q 1/ whenever Q 1 Q . The set of all natural numbers N ordered by the standard ordering Ä is a directed set. The nets having N (directed by this standard ordering) as the index set are called sequences. Every subsequence of a given sequence can be simultaneously understood as a finer net.2 A.2 Topology A topology T of a set X is a collection of subsets of X such that T contains the empty set and X itself, and with every finite collection of sets also their intersection, and also with every arbitrary collection of sets also their union. The elements of T are called open sets (or T-open, if we want to indicate explicitly the topology in question), while their complements are called closed.AsetX endowed with a topology T is called a topological space; sometimes we denote it by .X; T/ to refer to T explicitly. A collection T0 of subsets of X is called a base of a topology T if every T-open set is a union of elements of T0. Then T WD f [˛A˛ j8˛ W A˛ 2 T0 g is a topology (induced by the base T0). Having a subset A X, TjA WD f A \ B j B 2 T g is a topology on A, called a relative topology.Havingx 2 N X, we say that N is a neighborhood of x if there is an open set A such that x 2 A N;thesetofall neighborhoods of a point x is denoted by N .x/ or, more specifically, NT.x/ if the topology T needs to be specified. A topology T is called Hausdorff if 8x1; x2 2 X, x1 ¤ x2 9A1 2 N .x1/; A2 2 N .x2/: A1 \ A2 D;. We define the interior,theclosure, and the boundary of a set A respectively by n ˇ o ˇ int.A/ WD x 2 X ˇ 9N 2 N .x/ W N A ; (A.2.1a) 1This assertion is unfortunately highly nonconstructive unless X N, and is equivalent to the axiom ofS choice: for every set X and every collection fAxgx2X, ;¤Ax X, there is a mapping . / f W X ! x2X Ax such that f x 2 Ax for every x 2 X. 2 Indeed, having a sequence .xk/k2N and its subsequence .xk/k2N with some N N, one can put WD .N; Ä/, Q WD .N; Ä/,andj W Q ! the inclusion N N. A Elements of topology and functional analysis 581 n ˇ o ˇ cl.A/ WD x 2 X ˇ 8N 2 N .x/ W N \ A 6D; (A.2.1b) @A WD cl.A/ n int.A/: (A.2.1c) Having A B X, we say that A is dense in B if cl.A/ B. A topological space is called separable if it contains a countable subset that is dense in it. Having a net .x /2 in the topological space X, we say that it converges to a point x 2 X if for every neighborhood N of x, there is 0 2 large enough that x 2 N whenever 0; then we say also that x is the limit point of the net in question, and write lim2 x WD x or simply x ! x. This concept of convergence is called the Moore–Smith convergence [433]. Note that x 2 cl.A/ if and only if there is a net in A converging to x; in this case, we also say that x is attainable by a net from A. A point x 2 X is called a cluster point of the net .x /2 if for every neighborhood N of x and for every 0 2 , there is 0 such that x 2 N. Clearly, every limit point is a cluster point as well, but not conversely. Nevertheless, for every 3 . / cluster point x of a net x 2 , there exists a finer net fxQ Q gQ 2Q converging to x. Ordering of all topologies on a given set X is naturally by inclusion: having two topologies T1 and T2 onasetX, we say that T1 is finer than T2 or T2 is coarser than T1 if T1 T2 (or equivalently, if the identity on X is .T1; T2/-continuous). The adjectives “stronger” and “weaker” are sometimes used in place of “finer” and “coarser,” respectively. A function d W XX ! R1 is called a quasidistance on X if for all x1; x2; x3 2 X, d.x1; x2/ 0, d.x1; x2/ D 0 is equivalent to x1 D x2, and d.x1; x2/ Ä d.x1; x3/ C d.x3; x1/. A quasidistance that does not take values 1 is called a distance.Every distance d induces a topology T by a base ffx 2 X j d.x; x1/<"gjx1 2 X;">0g. A distance d W XX ! R is called a metric on X if d.x1; x2/ D d.x2; x1/ for all x1; x2 2 X. Conversely, a topology is called metrizable if there exists a metric that induces it. However, it should be emphasized that there exist nonmetrizable topologies. We say that a topological space is compact if every open cover admits a finite subcover. In terms of nets, the equivalent definition reads that every net has a cluster point. There are various useful modifications of this notion. We say that a topology is sequentially compact if every sequence in X admits a subsequence that converges in X. A metrizable topology is compact if and only if it is sequentially compact. A set A is called relatively (sequentially) compact if the closure of A is (sequentially) compact in X. A topological space is called locally (sequentially) compact if every point of it possesses a (sequentially) compact neighborhood. 3It suffices to put Q WD N .x/ directed by the ordering Äand to take, for every Q D .; / Q Q N 2 ,somexQ Q WD x 2 N with .
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