ADVANCE TOPICS IN TOPOLOGY - POINT-SET NOTES COMPILED BY KATO LA 19 January 2012 Background Intervals: pa; bq “ tx P R | a ă x ă bu ÓÓ , / / calc. notation set theory notation / / \Open" intervals / pa; 8q ./ / / p´8; bq / / / -/ ra; bs; ra; 8q: Closed pa; bs; ra; bq: Half-openzHalf-closed Open Sets: Includes all open intervals and union of open intervals. i.e., p0; 1q Y p3; 4q. Definition: A set A of real numbers is open if @ x P A; D an open interval contain- ing x which is a subset of A. Question: Is Q, the set of all rational numbers, an open set of R? 1 1 1 - No. Consider . No interval of the form ´ "; ` " is a subset of . We can 2 2 2 Q 2 ˆ ˙ ask a similar question in R . 2 Is R open in R ?- No, because any disk around any point in R will have points above and below that point of R. Date: Spring 2012. 1 2 NOTES COMPILED BY KATO LA Definition: A set is called closed if its complement is open. In R, p0; 1q is open and p´8; 0s Y r1; 8q is closed. R is open, thus Ø is closed. r0; 1q is not open or closed. In R, the set t0u is closed: its complement is p´8; 0q Y p0; 8q. In 2 R , is tp0; 0qu closed? - Yes. Chapter 2 - Topological Spaces & Continuous Functions Definition:A topology on a set X is a collection T of subsets of X satisfying: (1) Ø;X P T (2) The union of any number of sets in T is again, in the collection (3) The intersection of any finite number of sets in T , is again in T Alternative Definition: ¨ ¨ ¨ is a collection T of subsets of X such that Ø;X P T and T is closed under arbitrary unions and finite intersections. Definition: A set X for which a topology has been defined is called a topological space. Example: X “ ta; b; cu T1 “ tØ; X; tau ; ta; buu is a topology on X. T2 “ tØ;Xu is known as the \trivial" topology. T3 “ tall subsets of Xu is known as the \discrete" topology. T4 “ tØ; X; tb; cu ; tcuu has complements of sets in T1. Notice which Tx are subsets of Ty and vice versa. Definition: A set which belongs to T is called an open set. Example: If we let T contain all the sets which, in a calculus sense, we call open - We have \R with the standard [or usual] topology." Example: [Example 3, Page 77 in the text] X is a set. Tf contains all sets whose complements is either X or finite OR contains Ø and all sets whose complement is finite. Recall: pA X BqA “ AA Y BA and pA Y BqA “ AA X BA ADVANCE TOPICS IN TOPOLOGY - POINT-SET 3 Checking condition (2): Suppose tUαu is a collection of sets in Tf . A A Taking the complement of Uα ñ Uα “ Uα ñ which must be finite because A each of the Ui is finite. ¤ ´¤ ¯ £ ` ˘ n A n A A A Checking condition (3): Ui “ Ui each Ui is finite. ñ Ui must ˜i“1 ¸ i“1 be finite. £ ¤ ` ˘ ¤ ` ˘ Basis for a Topology Definition: If X is a set, a basis for a topology T on X is a collection B of subsets of X [called \basis elements"] such that: (1) Every x P X is in at least one set in B (2) If x P X and x P B1 X B2 [where B1;B2 are basis elements], then there is a basis element B3 such that x P B3 Ă B1 X B2 Question: How in fact do you know that you get a topology from basis elements? Examples: [of bases] (i) Open intervals of the form pa; bq are a basis for the standard topology on R. 2 (ii) Interior of circle are a basis for the standard topology in R . (iii) All one-point subsets of X are a basis for the discrete topology. 26 January 2012 Examples: (i) X “ ta; b; c; d; eu, T1 “ tØ; X; tau ; ta; buu, T2 “ tØ; X; tau ; tcu ; ta; bu ; ta; b; cu ; ta; cuu. T2 Ą T1, so T2 is “finer” than T1. i.e., The set that is “finer” has more open sets. We say, \tau is open in T1" which is equivalent to tau P T1. A In T2, tau is open. tau is not closed since tau “ tb; c; d; eu is not open. (ii) On the real number line, T is referred to as the \usual" or \standard" topology on R: (a) Sets in T are sets that are \open" in the calculus sense - i.e., p1; 3q; p0; 8q, and p1; 3q Y p4; 6q. (b) Sets in T are those that are unions of all sets of the from pa; bq where a ă b. 4 NOTES COMPILED BY KATO LA Recall the definition of a basis: A basis for a topology pX; T q is a collection of subsets B such that: (1) @ x P X is in some B P B (2) If x P X and x P B1 X B2, then there is B3 such that x P B3 Ă B1 X B2 Note: Every basis \generates" a topology this way: A subset U is \open" if every element in U satisfies: x P B Ă U for some B P B. Three Classic Lemmas Lemma 1: Every element in T is a union of basis elements. Lemma 2: If pX; T q is a topological space and C is a collection of open sets such that for every open set A, if x P A, then x P C Ă A for some C P C , then C is a basis. Lemma 3: Let B and B1 be bases for T and T 1 on X. Then T 1 is finer than T if and only if for each x and each B P B containing x, there is B1 P B1 such that x P B1 Ă B. Further examples: (i) R` is the \lower limit topology" on R where a basis is the set of all intervals of the form ra; bq where a ă b. Is (0, 1) open in R`? i.e., Can we write (0, 1) as a union of sets of the form ra; bq? 1 1 1 - Yes. (0, 1) = ; 1 Y ; 1 Y ¨ ¨ ¨ Y ; 1 Y ¨ ¨ ¨ 2 3 n „ ˙ „ ˙ „ ˙ How do R` and R compare? i.e., Can we write [0, 1) as a union of sets of the form pa; bq? - No. Consider using the same method as above. If we attempt to include zero, say 1 ´ ; 1 Y ¨ ¨ ¨ , will include zero, but as well as erroneous values less than zero. 100 „ ˙ Every set open in R is open in R`, but not the converse. Thus, R` is finer than R. ADVANCE TOPICS IN TOPOLOGY - POINT-SET 5 (ii) Rk is the so-called \k-topology" where the basis elements are intervals of the form 1 1 1 pa; bq OR pa; bq z k where k “ 1; ; ; ; ¨ ¨ ¨ . How do and compare? 2 3 4 R Rk " * Well, all sets open in R are open in Rk. Are sets open in Rk open in R as well? ? 1 1 1 1 1 i.e., p´1; 1q z k “ p´1; 0q Y ; 1 Y ; Y ; Y ¨ ¨ ¨ . 2 3 2 4 3 ˆ ˙ ˆ ˙ ˆ ˙ This is very close because 0 P left-hand side, but 0 R right-hand side. So Rk is finer than R. How do R` and Rk compare? 1 1 r0; 1q ‰ ; 1 Y ; 1 Y ¨ ¨ ¨ . 2 3 ˆ ˙ ˆ ˙ r0; 1q is open in R`, but not in Rk and p´1; 1q is open in Rk, but not in R`. So R` and Rk are not comparable. i.e., Just like skew lines! A topology on Z: tnu if n is odd Open sets in the basis have two forms: tn ´ 1; n; n ` 1u if n is even " This is the \Digital Line Topology." Is p3; 6q open? - No. p3; 6q “ t3u Y t3; 4; 5u Y t5u Y t5; 6; 7u. In fact, is any set like t6u [\even," in other words] open? - No. But r3; 7s is open: r3; 7s “ t3; 4; 5u Y t5; 6; 7u Note: t3u and t5u are open sets in the basis, but not necessary as they are included in r3; 7s by t3; 4; 5u and t5; 6; 7u respectively in the digital line topology. 6 NOTES COMPILED BY KATO LA 2 February 2012 Basis ñ Topology If we have a basis, we get a topology this way: Definition - Basic Property [BP]: A set A is called open if every element in A is in some basis element which is a subset of A. i.e., Definition of open when given a basis. Note: If B is a basis element, is B open? - Yes! If x P B, then x P B Ă B. basis element Do all sets U that satisfy the basic property form a topology? ljhn (1) Does Ø satisfy BP? - Yes, vacuously. Does X satisfy BP? i.e., If x P X, is there a B so that x P B Ă X?- Yes, by definition of a basis. (2) If tUαu is a collection of sets satisfying BP, does Uα? Since all Uα satisfy BP, α ¤ for each Uα and x P Uα, there is a B such that x P B Ă Uα.