Download MATH 358 – FINAL EXAM REVIEW The following is

MATH 358 – FINAL EXAM REVIEW
The following is designed to help you prepare for the final exam. A substantial portion of the final will
be derived from these questions and their close relatives.
1. What does it mean?
Give definitions of each of the following. Where appropriate, provide a short, illustrative example.
• The preimage of a set U under function f . . .
• The image of function f . . .
• The graph of a function f : X → Y . . .
• A set X is countable. . .
• A relation on X is an order relation. . .
• A topology on set X. . .
• A basis on set X. . .
• The topology generated by a basis. . .
• The order topology. . .
• The product topology. . .
• The subspace topology. . .
• The quotient topology. . .
• Topology T1 on set X is finer than topology T2 . . .
• A function f : X → Y is continuous. . .
• A subset A of topological space X is closed . . .
• Let A be a subset of X. The interior int(A). . .
• Let A be a subset of X. The closure A. . .
• The standard topology on R. . .
• The lower-limit topology on R. . .
• The finite-complement topology on R. . .
• The dictionary order on X × Y . . .
• The dictionary order on Z2 . . .
• A function f : X → Y is a homeomorphism. . .
• A topological space X is connected . . .
• A topological space X is path-connected . . .
• A topological space X is Hausdorff . . .
• A topological space X is compact. . .
• A subset K of topological space X is compact. . .
• A homotopy between functions f and g. . .
• A path homotopy. . .
• A set K in Rn is convex . . .
• A topological space X is simply connected . . .
Date: Fall 2013 .
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MATH 358 – FINAL EXAM REVIEW
• The fundamental group π1 (X) of a topological space X. . .
• A lift of f : [a, b] → S 1 . . .
• The winding number of f : [a, b] → S 1 . . .
• A retract A of X. . .
2. True or False?
Determine if each of the following statements are true of false. If a statement is true, give a short sketch
of the proof; if false, provide a counter-example. All sets X, Y , etc. are assumed to be topological spaces
unless otherwise specified.
(1) The set of rationals is countable.
(2) The set {3, 4, 5}ω is countable.
(3) Countable unions of countable sets are countable.
(4) The relation R on Z+ given by aRb iff a2 < b is an order relation.
(5) If each Uα are open in topological space X for α in index set J, then ∪α Uα is open in X.
(6) If each Uα are closed in topological space X for α in index set J, then ∪α Uα is closed in X.
(7) If each Uα are open in topological space X for α in index set J, then ∩α Uα is open in X.
(8) If each Uα are closed in topological space X for α in index set J, then ∩α Uα is closed in X.
(9) There exists a largest topology on set X.
(10) There exists a smallest topology on set X.
(11) Let X be a topological space and give Y ⊂ X the subset topology. If U ⊂ Y is open in Y , then it is
also open in X.
(12)
(13) {(a, b) × (c, d) | a, b, c, d ∈ R} is a basis for the standard topology on R2 .
(14) For x ∈ R2 and r ∈ (0, ∞), define Br (x) = {y ∈ R2 | kx − yk < r}. The collection B = {Br (x)}
forms a basis for the standard topology on R2 .
(15) The collection {U ⊂ X | x \ U is empty or infinite or all of X} is a topology on X.
(16) If Tz is a topology on X for each z ∈ R, then ∩z Tz is also a topology on X.
(17) If Tz is a topology on X for each z ∈ R, then ∪z Tz is also a topology on X.
(18) The set U = ∪k∈Z (k − 71 , k + 17 ) is open in Rl .
(19) (1, 2) × (0, 1) ∪ (3, 4) × (0, 1) is an element of the basis for the product topology on R × R.
(20) All functions are continuous.
(21) If f : X → Y is continuous, then f (X) is open.
(22) If f : X → Y is continuous, then f (X) is not open.
(23) If f : X → Y is continuous, then f (X) is closed.
(24) If f : X → Y is continuous, then f −1 (Y ) is open.
(25) If f : X → Y is continuous, then f −1 (Y ) is not open.
(26) If f : X → Y is continuous, then f −1 (Y ) is closed.
(27) If the graph of f : X → Y is closed, then f is continuous.
(28) If f : X → Y is continuous, then the graph of f is closed.
(29) If f : X → Y is a continuous bijection, then X and Y are homeomorphic.
(30) Homeomorphism is an equivalence relation on the set of topological spaces.
(31) Constant functions are continuous.
(32) If A ⊂ X has the subspace topology then the embedding j : A → X is continuous.
MATH 358 – FINAL EXAM REVIEW
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(33) The functions f : X → Y and g : Y → Z are continuous if and only if the composite f ◦ g : X → Z
is continuous.
(34) The integers are connected with respect to the topology induced by Rstd .
(35) The rationals are connected with respect to the topology induced by Rstd .
(36) The graph of a function is connected (with respect to the subspace topology) if and only if the
function is continuous.
(37) X × Y is connected if and only if X and Y are connected.
(38) The intervals are the only bounded connected subsets of Rstd .
(39) If a space is path-connected, then it is connected.
(40) If a space is connected, then it is path-connected.
(41) If X is Hausdorff and A ⊂ X is finite, then A is closed.
(42) If X is Hausdorff then X is connected.
(43) Closed subspaces of compact spaces are compact.
(44) If K is a compact subset of Hausdorff space X and y ∈ X \ K then there exists open U, V with
K ⊂ U , y ∈ V , and U ∩ V = ∅.
(45) Any topological space contains at least one compact subset.
(46) Any interval of Rstd contains a non-compact subset.
(47) If Y ⊂ X has the subspace topology and K is a compact subset of Y , then K is also a compact
subset of X.
(48) If Kj are compact subsets of Rstd for j ∈ Z+ , then ∪j∈Z+ Kk is also compact.
(49) If Kj are compact subsets of Rstd for j ∈ Z+ , then ∩j∈Z+ Kk is also compact.
(50) If K ⊂ R is compact with respect to the standard topology, then it is also compact with respect to
the lower-limit topology.
(51) If f : X → Y is continuous and X is connected, then so is f (X).
(52) If f : X → Y is continuous and Y is connected, then so is f −1 (Y ).
(53) If f : X → Y is continuous and X is path-connected, then so is f (X).
(54) If f : X → Y is continuous and Y is path-connected, then so is f −1 (Y ).
(55) If f : X → Y is continuous and X is compact, then so is f (X).
(56) If f : X → Y is continuous and Y is compact, then so is f −1 (Y ).
(57) Let f : X → Y be a continuous bijection. Then X is compact if and only if Y is.
(58) Path homotopy is an equivalence relation on the collection of all paths from point P to point Q.
(59) If f : K → X is continuous and K ⊂ Rn is convex, then f is homotopy equivalent to a constant
function.
(60) If f : K → X is continuous and K ⊂ Rn is compact, then f is homotopy equivalent to a constant
function.
(61) All convex sets are simply connected.
(62) All simply connected sets are convex.
(63) If X \ {∗} is simply connected, then so is X.
(64) Suppose f : [0, 1] → S 1 has a winding number of zero. Then f is not surjective.
(65) Suppose f : [0, 1] → S 1 is injective. Then the winding number of f is zero.
(66) Let n ≥ 0 and suppose f : [0, 1] → S n is not surjective. Then f is nullhomotopic.
(67) S 1 is a retract of R2 .
(68) S 1 is a retract of R2 \ {∗}.
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MATH 358 – FINAL EXAM REVIEW
(69) S 1 is a retract of T 2 .
(70) S 1 is a retract of T 2 \ {∗}.
(71) S 1 is a retract of S 2 .
(72) S 1 is a retract of S 2 \ {∗}.
(73) A space X is simply connected if and only if S 1 is a retract of X.
(74) Connectedness is a homeomorphism invariant.
(75) Path-connectedness is a homeomorphism invariant.
(76) Compactness is a homeomorphism invariant.
(77) Simple connectedness is a homeomorphism invariant.
(78) Suppose X and Y are compact and simply connected. Then X ∼
=Y.
3. Examples
I encourage you to make a long list of examples, and types of examples, we have encountered. You should
be able to:
• Give the basic properties of each space (compact, connected, etc.)
• Give the basic properties of subsets of each space.
• Determine which spaces are homeomorphic and, if they are, present (sketches of) homeomorphisms.
• Determine which spaces are not homeomorphic and identify a distinguishing characteristic or property.