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L100601 - HA 55-5 Topology llly'lflll'la'r DEFA elnaed not...

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Unformatted text preview: HA. 55-5 Topology - llly'lflll'la'r DEFA elnaed not in. the complement of Home open aet — i.e. [I in. closed iffi' = I — H, where I] in open. DEFA lln1lt point of a net A it a pointII a E I, 1I.rith the following property: 1] Any open not which centaine 2 contains some point of A. Notice that if r E A, then r ia a. limit point oi' all. LEIII A eat ia cloned if it eentaina all of ita limit pointa. Ff.li'i"hat pointe are not limit pointt of A? a in not a limit point of .11 iii there in- an open aet U. ouch that film A = fl 4:: E", E I — .11. The net ofnon-ljmit pointe efii. ie open. In a metric space, a limit point el A in a point e E I such that there are points in A arbitrarily clone to 1. An alternative way of specifying a topology on a net it in to specify the cloned aeta. Eaaic Properties ot'eloaecl aeta i} Arbitrary interaeetiena of closed aata are cloaed. I.e. if {FE} ia a. collection of cloned BetaT then FIFE ia cloned. {I — HF... = UU'II' — Fer} and I — F... in open.) 1|} Finite union of cloned nets are closed. F15Fl.-..Fn open ==~ ULIFr open. ill]|fl.I are cloned EILIn I. lIauedorE epace, a. one-point eat,1[=]- ie cloned. Pf.We need to chow I — {r} ia open. Let y = I — {1r}. Then 1are can find open set: V, and 3;. containing :1 and r retpectiyely. FurthermoreI I — {a} = Ufix—{=}Vfr U!“ I"! = H =5 Iill-1| E I — Er” E I — {5} In the indiecrete topology, the only closed eete are H and I. ED H.111 a Haucdorff space, any finite Bet 111 cloced. EILEA etrange topology - the eoflnlte topology] Clot-er] nets are iinite Beta and I. [Thie eatiafiee the propertiea of eloaed aete liatell above.) RE” .The eofinita topolgy ia not EeuadorE. Pf. Bnppoae that we have UTV. ii" at '3' to U = I — F1 , 1where F in finite. Similarly V. I —[F'1U.F:;l= [I —F1}I'II:X —.F':j= Fn‘lr' = ii. 3111,3- —{F1UFIJ 5‘é a. ltll'clmr way: of conclruering topologies - been and when“ DEFH we are given a topological apnea {1.1.}, F E 'T in eairl to he a haaa for T if any I? E T can be written as a. union of late in B. E1.HEI.pllB a metric space. H = {B{r,r]| :r. E 3.1- :5 II] in a hate for the topology generated by the metric. Recall that we defined open rot: in a metric opaee to be arbitrary unions of hallo. l ...
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