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NOTES WEEK 08 DAY 1 SCOT ADAMS Let T be a set of sets. We recalled that T is a topology means: xT yY “ T “ xT yfin X . Also, for any set X, T is a topology on X Ť means: ( T is a topology ) and ( T “ X ). Also, for any set X, pX, T q is a topological space means: T is a topology on X. Let pX, T q be a topological space and let A Ď X. Recall that, for any x P X, by A is a nbd of x, we mean: DU P T s.t. x P U Ď A. Recall: @set S of sets, @set A, we defined S|A :“ tS X A | S P Su. Let pX, T q be a topological space and let A Ď X. Then the relative topology on A inherited from pX, T q is T |A. (It is an unassigned exercise to show that T |A is a topology on A.) NOTE TO SELF: Next time I teach this, be sure to go over: ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ Every Every Every Every Every Every Every Every subspace open extends to an ambient open. subspace closed extends to an ambient closed. subspace nbd extends to an ambient nbd. subspace pnbd extends to an ambient pnbd. ambient open restricts to a subspace open. ambient closed restricts to a subspace closed. ambient nbd restricts to a subspace nbd. ambient pnbd restricts to a subspace pnbd. Recall that: BR :“ tpa, bq | a, b P R, a ă bu, B˚ :“ BR Y tr´8, bq | b P Ru Y tpa, 8s | a P Ru, TR :“ xBR yY , T˚ :“ xB˚ yY . and These are: the standard base on R, the standard base on R˚ , the standard topology on R and the standard topology on R˚ . REMARK 0.1. Let S be a set of sets and let A be a set. Then we have pxSyY q|A “ xS|AyY and pxSyX q|A “ xS|AyX . Also, we have pxSyfin Y q|A “ xS|Ayfin Y and pxSyfin X q|A “ xS|Ayfin X . Date: October 25, 2016 Printout date: November 28, 2016. 1 2 SCOT ADAMS Proof. Omitted. COROLLARY 0.2. Let pX, T q be a topological space and let B be a base for T . Let A Ď X. Then B|A is a base for T |A. Proof. Because B is a base for T , we conclude that xByY “ T . Then xB|AyY “ pxByY q|A “ T |A. Thus B|A is a base for T |A, as desired. DEFINITION 0.3. For any A Ď R˚ , we define BA :“ B˚ |A and TA :“ TA . These are called the standard base on A and standard topology on A, respectively. DEFINITION 0.4. We define N˚ :“ N Y t8u. REMARK 0.5. Let A :“ t100, 101, 102, . . .uYt8u. Then A is a basic open nbd of 8 in N˚ . That is, using the standard base and topology on N˚ , we assert that A is a nbd of 8 and that A is basic open. (Recall that “basic open” means “an element of the base”.) Proof. Since p99, 8s is a basic open subset of R˚ , it follows that p99, 8sX N˚ is a basic open subset of N˚ . So, since A “ p99, 8s X N˚ , we see that A is a basic open subset of N˚ . So, since 8 P A. It remains to show that A is a neighborhood of 8 in N˚ . Since 8 P A and since A is open in N˚ , we conclude that A is a neighborhood of 8 in N˚ . Note, also that t98u Y t100, 101, 102, . . .u Y t8u is an open nbd of 8 in N˚ , but is not basic open. DEFINITION 0.6. Let pX, T q be a topological space and let a P X. Let N be a set of nbds of a. Then N is a nbd base at a in pX, T q means: @nbd S of a in pX, T q, DB P N s.t. B Ď S. We often either abbreviate “in pX, T q” to “in X”, or simply omit “in pX, T q”, if the context is sufficiently clear. Some examples: ‚ Let a P R. Then tpa ´ ε, a ` εq | ε ą 0u is a nbd base at a both in R and in R˚ . This nbd base is called the standard nbd base at a. ‚ Let a P R. Then tpa ´ p1{nq, a ` p1{nqq | n P Nu is a countable nbd base at a both in R and in R˚ . This nbd base is called the standard countable nbd base at a. NOTES WEEK 08 DAY 1 3 ‚ The set tpM, 8s | M P Ru is a nbd base at 8 in R˚ . This nbd base is called the standard nbd base at 8. ‚ The set tpM, 8s | M P Nu is a countable nbd base at 8 in R˚ . This nbd base is called the standard countable nbd base at 8. ‚ The set tr´8, ´M q | M P Ru is a nbd base at ´8 in R˚ . This nbd base is called the standard nbd base at ´8. ‚ The set tr´8, ´M q | M P Nu is a countable nbd base at ´8 in R˚ . This nbd base is called the standard countable nbd base at ´8. ‚ Let pX, dq, for every a P X, Then tBpa, rq | r ą 0u is a nbd base at x in pX, Td q. This nbd base is called the standard nbd base at x. ‚ Let pX, dq, for every a P X, Then tBpa, 1{nq | n P Nu is a countable nbd base at x in pX, Td q. This nbd base is called the standard countable nbd base at x. DEFINITION 0.7. For any set S, for any a, we define Saˆ :“ Sztau. DEFINITION 0.8. Let pX, T q be a topological space, let P Ď X and let a P X. Then P is a T -pnbd or, simply pnbd of a means: there exists a neighborhood S of a in pX, T q such that P “ Saˆ . In Definition 0.8, “pnbd” is short for “punctured neighborhood”. For example, t100, 101, 102, . . .u is a pnbd of 8 in N˚ . NOTE TO SELF: Next time, in Definition 0.9 below, say “for every nbd V of b in pimrf sqYtbu, there exists a pnbd P of a in pdomrf sqYtau such that f˚ pP q Ď V ”. Then follow up with an explanation (HW?) that it’s equivalent to: “for every nbd V of b in Y , there exists a pnbd P of a in X such that f˚ pP q Ď V ”. DEFINITION 0.9. Let pX, T q and pY, Uq be topological spaces, let f : X 99K Y and let a P X. Then, for all b P Y , the notation ( f Ñ b near a ) means: for any nbd V of b in Y , there exists a pnbd U of a in X such that f˚ pP q Ď V . Also, LIMSpf q :“ tb P Y | f Ñ b near au. a Also, lim f :“ ELTpLIMSpf qq. a a When x is unbound, “f pxq Ñ b as x Ñ a” means “f Ñ b near a”, with temporary binding on x. When t is unbound, “f ptq Ñ b as t Ñ a” means “f Ñ b near a”, with temporary binding on t. Etc. 4 SCOT ADAMS When x is unbound, “lim f pxq” means “lim f ”, with temporary xÑa a binding on x. When t is unbound, “lim f ptq” means “lim f ”, with tÑa a temporary binding on t. Etc. Example: Define a P RN by aj “ 1{j. Then a : N˚ 99K R˚ . We played a game to show: ( a‚ Ñ 0 near 8 ). We played a game to show: NOT ( a‚ Ñ 1 near 8 ). We argued that LIMSpa‚ q “ t0u. We con8 cluded that lim a‚ “ 0, or, equivalently, lim aj “ 0. More generally, jÑ8 8 we have: FACT 0.10. Let a P RN . Then, for every z P R˚ , we have: p a‚ Ñ z q ô p a‚ Ñ z near 8 q. Also, LIMSpa‚ q “ LIMSpa‚ q. Also, lim a‚ “ lim a‚ . 8 Proof. Omitted. 8