Question

It's given the following metric space [ir?,d) with the euclidean norm and the set M= {Ixoy)ER?: Osxs1, Osyste} show or disprove that a) M is a closed set b) M is a bounded set c) M is compact

Answer 1

nonm where prove M Given (18², d) be a metric space with the enclidean Then din,y) = N(Ra-Y, P+(12-9,12 ka (182) €1R2 and y= (4., Y₂) E 1R2 How the given sel, Ma2 (2,4) EIR? : OSK SA, osya sa }, Now (a) we is dosed by sequential criterion. let. Boros Xn=(kn. Yn) such that som fxn's nas sequence on M and and d(xn, xo) o eventually, where to (no. Yo). XnEM SO o sansa and osa Yas Ikan be AS An EIN Now , (xn, xo) to z wcXm-20)+(yn-Y) 2 eventually eventually eventually » (Xn-ko)² + (Yn-Yol? knoodo and Yuwoyo eventually (As the squre of eventually and Oskus 1 vnen there fore okost Again as Yn Yo eventually and o syns han unein therefore os yo s taco Thus for every sequence {xms in M and d(xn, xo) to eventually implies to EM. two term goes to o]. As anno

osn'sa and Therefore sequential Griterion of closed set M is closed. (b) Given, Ma {(2,4) € R?: Osve sto systx}, Now let (X', Y') EM s osa's1 and of y's Vixi osk'st and osy's sal e I = L. os y's (0,0) 3 (very') 3 (1,1). Thus for all (ny) EM a 10,0) & (2, 4){ (1,1) Therefore M is bounded. (C) we get from casM is closed and from (b) M is bounded. Therefore the Heine-Bonel theorem for euclidean space R2 m' is compact. any closed bounded set in euclidean space 22 is compuet. So M is Becouse word compact,