Math 205A Homework #2 Edward Burkard. Assume each composition with a projection is continuous. Let U Y Y be an open set.

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1 Math 205A Homework #2 Edward Burkard Problem - Determne whether the topology T = fx;?; fcg ; fa; bg ; fa; b; cg ; fa; b; c; dgg s Hausdor. Choose the two ponts a; b 2 X. Snce there s no two dsjont open sets separatng them, T s not T 2. Y Problem 2 - For a collecton of spaces Y show that a functon f : X! Y s contnuous () each composton X! Y Y! Y, wth the projecton, s contnuous. (=)) Assume f s contnuous. Then snce projecton maps are contnuous the composton of f and a projecton map s contnuous. ((=) Assume each composton wth a projecton s contnuous. Let U Y Y be an open set. Snce (the projecton) s open 8, (U) s open 8, thus ( f) ( (U)) = f (U) = f (U) s open. Problem 3 - If X s a topologcal space, the "dagonal" of X X s the subspace = f(x; x) j x 2 Xg. Show that X s T 2 () s closed n X X. X s T 2 () Gven (u; v) 2 C there are open sets U; V X such that u 2 U, v 2 V, and (U V ) \ =?. () C s open n X X. () s closed n X X.

2 Problem 4 - Let R be endowed wth the standard topology, and consder R! = Y n2n R endowed wth the product topology. Show that the subset S of all sequences fx n g n2n that are zero after ntely many terms s dense n R!. De ne a metrc on S that nduces the subspace topology. A subset s dense n a set f ts ntersecton wth any open set s nonempty. Thus t su ces to show that S \ B 6=? for all bass elements B of R!. A generc bass element B of R! has the form B = U U 2 U 3 U n R R R, where n 2 N and U s open n R 8 n. Choose x 2 U for each n and consder the sequence (x ; x 2 ; x 3 ; :::; x n ; 0; 0; 0; :::) = x. Clearly x 2 S \ B and thus S \ B 6=?. Snce B was arbtrary and any open set s constructed from bass elements, S \ V 6=? for any open set V n R!. To de ne a metrc on S that nduces the subspace topology we can de ne a metrc on R! that nduces the product topology on R! and restrct ts doman to S. Consder the metrc d (x; y) = mn fjx yj ; g on R. Ths metrc nduces the standart topology on nr. Extend ths to a metrc on R! by de nng for x; y 2 R! d(x o ;y, D (x; y) = sup ). It s easly ver ed that ths s a metrc. Now verfy that ths metrc nduces the product topology on R!. Let U be open n the metrc topology on R! and let x 2 U; 9V open n the product topology such that x 2 V U. Choose an " ball B R! (x; ") U. Then choose N large enough that < ". Fnally, let V be the bass element N for the product topology V = (x "; x + ") (x n "; x n + ") R R. Clam: V B R! (x; ") Gven any y 2 R!, Thus d (x ; y ) N for N: d (x ; y ) d D (x; y) max ; : : : ; (x N ; y N ) ; : N N 2

3 If y 2 V, ths expresson s less than ", so that V B R! (x; "), as desred. Conversely, consder a bass element U = Y 2N U for the product topology, where U s open n R for = ; : : : ; n and U = R for all other ndces. Gven x 2 U, we can nd an open set V of the metrc topology such that x 2 V U. Choose an nterval (x " ; x + " ) n R centered about x and contaned n U for = ; : : : ; n ; choose each ". Then de ne n " o " = mn j = ; : : : ; n : Let y be a pont of B R! (x; "). Then for all d (x ; y ) D (x; y) < ": Now f = ; : : : ; n, then " ", so that d (x ; y ) < " ; t follows that jx y j < ". ) y 2 Y 2N U. Thus ths metrc nduces the product topology on R!. Problem 5 - Let X and Y be metrc spaces wth metrcs d X and d Y, respectvely. Let f : X! Y have the property that for every par of ponts x ; x 2 of X, d Y (f (x ) ; f (x 2 )) = d X (x ; x 2 ). Show that f s a homeomorphsm onto ts mage,.e. an embeddng. (It s called an sometrc embeddng of X n Y.) f s : Suppose f (x ) = f (x 2 ) : Then 0 = d Y (f (x ) ; f (x 2 )) = d X (x ; x 2 ) : =) x = x 2 : f s onto: 3

4 Clearly f s onto ts mage f (X). f s contnuous: Let B f(x) (f (x) ; ") be an " Then ball centered at f (x) n f (X). f B f(x) (f (x) ; ") = f (ff (y) 2 f (X) j d Y (f (x) ; f (y)) < "g) whch s open n X. ) f s contnuous = fy 2 X j d X (x; y) < "g = B X (x; ") f s contnuous: Let B X (x; ") be an " ball centered at x n X. Then f (B X (x; ")) = f (fy 2 X j d X (x; y) < "g) whch s open n f (X). ) f s contnuous ) f s a homeomorphsm. = ff (y) 2 f (X) j d Y (f (x) ; f (y)) < "g = B f(x) (f (x) ; ") Problem 6 - Show that R wth the lower lmt topology s dsconnected. Let x 2 R. Consder the dsjont open sets ( ; x); [x; ) 2 R l. They consttute a separaton of R at x snce ( ; x) [ [x; ) = R. Thus R l s dsconnected snce t has a separaton. Problem 7 - Show that the Cantor set s compact. 4

5 Let A 0 be the closed nterval [0; ] n R. Let A be the set obtaned from A 0 by deletng ts "mddle thrd" ; Let A2 be the set obtaned from A by deletng ts "mddle thrds" ; and 7 ; In general, de ne An by the equaton A n = A n [ k=0 +3k 3 n ; 2+3k 3 n. The ntersecton C = \ n2n A n s the Cantor set; t s a subspace of [0; ]. Each A n s closed snce ts complement (A n ) C = A n [ s the unon of open ntervals, and thus open. Then snce C = k=0! C +3k; 2+3k 3 n 3 n \ A n, C s closed. Snce C [0; ], a closed subset of R, C s compact (C s bounded because t sts nsde a bounded subset of R). k=0 Problem 8 - Show that f X s a regular topologcal space, every par of ponts have neghborhoods whose closures are dsjont. Snce X s regular, t s also Hausdor. Thus gven any x; y 2 X we can nd dsjont open sets U and V wth x 2 U and y 2 V. Consder the closed set C = X U. By the regularty of X we can nd dsjont open sets A and B wth x 2 A and C B. Snce A X B and snce X B s closed we have that x 2 A A X B. Now consder the closed set F = X V. By the regularty of X we can nd dsjont open sets D and E wth y 2 D and F E. Snce D X E and snce X E s closed we have that y 2 D D X E. 5

6 Observe the followng: A X B X C = X (X U) = U D X E X F = X (X V ) = V Snce U \ V =? we have that A \ D =?. Thus we have two neghborhoods A and D of x and y respectvely whose closures are dsjont. by Problem 9 - Let X and Y be metrc spaces. De ne a metrc on X Y d ((x ; y ) ; (x 2 ; y 2 )) = d X (x ; x 2 ) 2 + d Y (y ; y 2 ) 2 2 : Show that the topology nduced by ths metrc s the product topology. A bass for the topology on X s gven by B = fb X (x; ") j x 2 X and " > 0g. A bass for the topology on Y s gven by C = fb Y (y; ) j y 2 Y and > 0g. Thus a bass for XY s gven by B C = fb X (x; ") B Y (y; ) j x 2 X, y 2 Y, " > 0, and > Let D = fb XY ((x; y) ; ) j (x; y) 2 X Y and > 0g. To prove ths t su ces to show that these two bases generate the same topology. Pck a pont (x; y) 2 X Y. Pck ; " > 0, and pck the bass element B X (x; ") B Y (y; ) n B C. Ths can be "vsualzed" as a rectangle wth sdes of length " n the "X drecton" and length n the "Y drecton". Choose the bass element B XY ((x; y) ; ) n D where = mn f; "g. Then B XY ((x; y) ; ) B X (x; ") B Y (y; ). Ths can be vsualzed by vewng B XY ((x; y) ; ) as a crcle of radus beng nscrbed n the rectangle B X (x; ") B Y (y; ) : Now pck > 0, and pck the bass element B XY ((x; y) ; ) n D. Choose the bass element B X (x; ") B Y (y; ) n B C where = " = p 2. Then B X (x; ") B Y (y; ) B XY ((x; y) ; ). Thus, snce any bass element from one bass contans a bass element from the other bass, and the pont (x; y) beng arbtrary, these two bases 6

7 generate the same topology, and thus the above metrc nduces the product topology. Problem 0 - Show that a completely regular space s regular. Let X be a completely regular space. Then we have 8x 2 X and closed C X wth x =2 C, a map f : X! [0; ] where f (x) = 0 and f on C. Consder the open sets [0; ) and ( ; ] n [0; ]. Then 2 2 f [0; ) and f ( ; ] are both open snce f s contnuous, and moreover 2 2 they are dsjont snce [0; ) \ ( ; ] =?. Thus, snce x 2 f [0; ) and C f ( ; ] we have just found dsjont open sets separatng x and C. 2 7