Unformatted text preview: HA. 555 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 ofnonljmit
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. onepoint 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 Iill1 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] Cloter] 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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 Spring '08
 Ogle,C
 Topology

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