1st December 2004
Munkres 26
Ex. 26.1 (Morten Poulsen). (a). Let T and T be two topologies on the set X . Suppose T T . If (X, T ) is compact then (X, T ) is compact: Clear, since every open covering if (X, T ) is an open covering in (X, T ). If (X, T ) i
4th January 2005
Munkres 27
Ex. 27.1 (Morten Poulsen). Let A X be bounded from above by b X . For any a A is [a, b] compact. The set C = A [a, b] is closed in [a, b], hence compact, c.f. theorem 26.2. The inclusion map j : C X is continuous, c.f. theorem
1st December 2004
Munkres 28
Ex. 28.1 (Morten Poulsen). Let d denote the uniform metric. Choose c (0, 1]. Let A = cfw_0, c [0, 1] . Note that if a and b are distinct points in A then d(a, b) = c. For any x X the open ball Bd (x, c/3) has diameter less tha
April 21, 2006
Munkres 29
Ex. 29.1. Closed intervals [a, b] Q in Q are not compact for they are not even sequentially compact [Thm 28.2]. It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact. To
1st December 2004
Munkres 30
Ex. 30.3 (Morten Poulsen). Let X be second-countable and let A be an uncountable subset of X . Suppose only countably many points of A are limit points of A and let A0 A be the countable set of limit points. For each x A A0 th
1st December 2004
Munkres 31
Ex. 31.1 (Morten Poulsen). Let a and b be distinct points of X . Note that X is Hausdor, since X is regular. Thus there exists disjoint open sets A and B such that a A and b B . By lemma 31.1(a) there exists open sets U and V
1st December 2004
Munkres 32
Ex. 32.1. Let Y be a closed subspace of the normal space X . Then Y is Hausdor [Thm 17.11]. Let A and B be disjoint closed subspaces of Y . Since A and B are closed also in X , they can be separated in X by disjoint open sets
9th June 2005
Munkres 33
Ex. 33.1 (Morten Poulsen). Let r [0, 1]. Recall from the proof of the Urysohn lemma that if p < q then U p Uq . Furthermore, recall that Uq = if q < 0 and Up = X if p > 1. Claim 1. f 1 (cfw_r) =
p>r
Up
q <r
Uq , p, q Q.
Proof. By
1st December 2004
Munkres 34
Ex. 34.1. We are looking for a non-regular Hausdor space. By Example 1 p. 197, RK [p. 82] is such a space. Indeed, RK is Hausdor for the topology is ner than the standard topology [Lemma 13.4]. RK is 2nd countable for the sets
1st December 2004
Munkres 35
Ex. 35.3. Let X be a metrizable topological space. (i) (ii): (We prove the contrapositive.) Let d be any metric on X and : X R be an unbounded real-valued function on X . Then d(x, y ) = d(x, y ) + |(x) (y )| is an unbounded m
1st December 2004
Munkres 36
Ex. 36.1. Any locally euclidean space is locally compact (as open subspaces of euclidean space are locally compact [Cor 29.3]). A manifold is locally euclidean and Hausdor, so it is locally compact Hausdor, hence regular [Ex 3
2nd April 2005
Munkres 38
Ex. 38.4. Let X X be the StoneCech compactication and X cX an arbitrary com pactication of the completely regular space X . By the universal property of the StoneCech compactication, the map X cX extends uniquely / cX XB = BB zz
11th December 2004
Munkres 25
Ex. 25.1. R is totally disconnected [Ex 23.7]; its components and path components [Thm 25.5] are points. The only continuous maps f : R R are the constant maps as continuous maps on connected spaces have connected images. Ex.
1st December 2004
Munkres 24
Ex. 24.2 (Morten Poulsen). Let f : S 1 R be a continuous map. Dene g : S 1 R by g (s) = f (s) f (s). Clearly g is continuous. Furthermore g (s) = f (s) f (s) = (f (s) f (s) = g (s), i.e. g is an odd map. By the Intermediate Va
Kaley Burke Quiz 2.3
1. Find the zeros of the f ( x)
x2
6 x 13 in two ways:
a. Use the Quadratic Formula.
x x x x x
6 6 6 6
( 6) 2 2 36 52 2 16 2 4i v
(4)(1)(13)
2 3 2i
x
3 2i
b. Complete the Square
(Be sure to simplify your answer.)
13 4
Kaley Burke Quiz 2.4
1. For the function f ( x) a. Find the vertex.
2x 2
24x 32
b 2a
( 24) 2( 2) 24 6 4
*The vertex is at (-6, 40)
f ( 6) f ( 6) f ( 6)
2( 6) 2 24( 6) 32 72 144 32 40
b. Determine if there is a minimum or a maximum and state wh
Kaley Burke Quiz 2.5
1. Solve and check:
x
x 5
5
x 5 ( x 5) x 5 30 3 x x 9
2
x
5 5) 2 25
( x
x 10 x 10 x
2. Solve:
1 4 x 12
1 4(x 3) (x (x 3) 3) x
1 x
2
5 9
1 3)(x
x 3
5 3) 4 4 (x (x 3) 1 3)(x 3) 3) 4(x 4(x 3) 3) 5 (x 3)
(x
1 4(x 3)
3
Kaley Burke Quiz 1.7
1. Test xy x 2 3 algebraically for whether the graph is symmetric with respect to the x-axis, the y-axis, and then the origin give work and an answer for each. *x-axis:
xy ( x) 2 xy x
2
3
3
*y-axis: xy *origin:
x2
3
3
(
Kaley Burke Quiz 2.1
1. The speed of an Amtrak passenger train is 12 mph faster than the speed of a Central Railway freight train. The passenger train travels 400 miles in the same time it takes the freight train to travel 320 miles. Find the speed o
1st December 2004
Munkres 13
Ex. 13.1 (Morten Poulsen). Let (X, T ) be a topological space and A X . The following are equivalent: (i) A T . (ii) x A Ux T : x Ux A. Proof. (i) (ii): If x A then x A A and A T . (ii) (i): A = xA Ux , hence A T . Ex. 13.4 (M
1st December 2004
Munkres 16
Ex. 16.1 (Morten Poulsen). Let (X, T ) be a topological space, (Y, TY ) be a subspace and let AY. Y X Let TA be the subspace topology on A as a subset of Y and let TA be the subspace topology on A as a subset of X . Since
Y U
1st December 2004
Munkres 17
Ex. 17.3. A B is closed because its complement (X Y ) (A B ) = (X A) Y X (Y B ) is open in the product topology. Ex. 17.6. (a). If A B , then all limit points of A are also limit points of B , so [Thm 17.6] A B . (b). Since A
September 29, 2008
Munkres 18
Ex. 18.1 (Morten Poulsen). Recall the - -denition of continuity: A function f : R R is said to be continuous if a R R+ R+ x R : |x a| < |f (x) f (a)| < . Let T be the standard topology on R generated by the open intervals. Th
1st December 2004
Munkres 19
Ex. 19.7. Any nonempty basis open set in the product topology contains an element from R , cf. Example 7p. 151. Therefore R = R in the product topology. (R is dense [Denition p. 191] in R with the product topology.) Let (xi )
1st December 2004
Munkres 20
Ex. 20.5. Consider R with the uniform topology and let d be the uniform metric. Let C R be the set of sequences that converge to 0. Then R = C. : Since clearly R C it is enough to show that C is closed. Let (xn ) R C be a sequ
4th January 2005
Munkres 22
Ex. 22.2. (a). The map p : X Y is continuous. Let U be a subspace of Y such that p1 (U ) X is open. Then f 1 (p1 (U ) = (pf )1 (U ) = id1 (U ) = U Y is open because f is continuous. Thus p : X Y is a quotient map. (b). The map
27th January 2005
Munkres 23
Ex. 23.1. Any separation X = U V of (X, T ) is also a separation of (X, T ). This means that (X, T ) is disconnected (X, T ) is disconnected or, equivalently, (X, T ) is connected (X, T ) is disconnected when T T . Ex. 23.2. U
Performance Measures:
A performance measure/indicator is an objective measurement of performance against a target
or required level of performance established for a particular control measure. These measures
should lead to corrective action(s) to be taken