By Andrew Browder
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Extra resources for Mathematical Analysis: An Introduction
Eight. allow X be a tupological area, Y a Hausdorff house, and enable I : X -+ Y and nine : X -+ Y be non-stop. exhibit that {x EX: I(x) = g(x)} is closed. Hl'nce if I(x) = g(x) for all x in a dense subset of X, then 1= g. 6. eight workouts 151 nine. enable X be a metric house. If x E X and E eX, outline the space from x to E as p(x, E) = inf{p(x,y): y E E}. (a) express that the functionality x ....... p(x, E) is uniformly non-stop on X, for every E c X, and that E = {x: p(x, E) = o}. (b) Deduce from (a) that if F is a closed subset of X, there exist open units Gn (n E N) such that F = n~=IGn' (c) If E and F are disjoint closed subsets of X, express that /, outlined by means of /(x) _ p(x, E) - p(x, E) + p(x, F) , is a continuing real-valued functionality on X, with zero ~ / ~ 1, and E = /-1 ( {o} ), F = I ( {I} ). Deduce that there exist disjoint open units U and V with E c U and Fe V. r 10. enable X be a topological area, and allow / : X --+ R. exhibit that the set C of all issues x E X such that / is continuing at x is the intersection of a countable kin of open subsets of X. eleven. permit X and Y be topological areas, and enable / : X --+ Y. express that / is continuing if and provided that /(E) c /(E) for each E C X, if and provided that / I(E) c /-I(E) for each E c Y. 12. enable X be a topological house, and allow / : X --+ [-00, +00). we are saying that / is higher semicontinuous, abbreviated u. s. c. , if / is constant whilst R U { -oo} is given the topology 5"u of instance 6. four. equally, a mapping / of X into R U {+oo} is termed decrease semicontinuous, or l. s. c. , whether it is non-stop while the objective house has the topology $i. (a) exhibit that / : X --+ R is continuing (R having its traditional topology) if and provided that / is either u. s. c. and l. s. c. (b) express that if / and nine are u. s. c. , then >. > o. eighty are / + nine and >. /, for any genuine (c) convey that if $ is any choice of u. s. c. services on X, and nine is outlined by way of g(x) = inf{f(x) : / E $}, then nine is U. S. c. (d) convey that / is l. s. c. if and provided that - / is u. s. c. , and deduce the analogues of (b) and (c) for l. s. c. services. thirteen. a suite $ of subsets of a suite X is related to have the finite intersection estate ifn;=1 Fj "" zero for any finite subcollection {FI ,F2 , •. • ,Fn } of $. convey topological house X is compact if and provided that for each assortment {Fo : () E A} of cI08ed subsets of X which has the finite intersection estate, we've got n oEA Fo "" zero. 152 6. Topology 14. express that if E is a wholly bounded subset of a metric house X, then for each { > zero there exists a finite subset {Xl, . •• , Xn} of E such that E C U~=I B(Xk. {). 15. enable (X, 07) be a compact Hausdorff house. enable 07' be a topology on X that's strictly superior than 07, and enable 07" be a topology on X that's strictly weaker than . 0/. exhibit that (X, :Y') is Hausdorff hut now not compact, whereas (X, . 0/") is compact, yet no longer Hausdorff. sixteen. permit X be a compact Hausdorff area, and enable I : X -+ R. The graph of I is the set CU) = {(x, I(x» : x E X} C X x R. exhibit that I is constant if and provided that C(f) is compact. 17. allow X be a compact metric house, and that i : X -+ X an isometry, i.