Topology

Test File

Summer 1992

 

1.) Determine if the following are topologies for the given set. Prove your answer.

a.) X, T where T is the intersection of a collection of topologies for X.

b.) X, Let A be a subset of X and let S be a topology for X. T = {U (V A) U, V S}

c.) X is an infinite set, T = {U X U = or U is infinite}

2.) Prove the following.

a.) Let X be a finite set. The discrete topology and the cofinite topology are the same.

b.) Let f be a one-to-one function from a topological space (X, T) to a set Y with the discrete topology. The function, f, is continuous if and only if T is the discrete topology.

c.) Let X be a set with more than one element and having the discrete topology. Which sets, if any, are dense?

1.) Define the following terms.

a.) Topology b.) Closed Set c.) Closure of a Set

d.) Dense Set e.) Limit Point f.) T1 - T2 continuous


2.) True-False - Write "True" only if the statement is always true. Otherwise, write "False" and give an example to show that it is false.

a.) In a topological space, (X, T), all subsets of X are either open or closed.

b.) In a topological space, (X, T), open sets can also be closed.

c.) In a topological space, the closure of a set is closed.

d.) Every topology contains at least 2 distinct open sets.

e.) [0, 1) is a closed set in (R, H).

f.) In a topological space, (X, T), a union of open sets is open.

g.) In a topological space, a union of closed sets is open.

3.) Determine if the following are topologies for the given set.

a.) X, T = {U X U = or A U, where A is a fixed subset of X}

b.) X = Y Z, where (Y, T1) and (Z, T2) are topological spaces. T = {U V U T1, V T2}

4.) Prove the following.

a.) A set, A, is closed if and only if for every point, x, not in A, there exists an open set, Ux, such that x Ux X - A.

b.) Let f be a function from a topological space (X, T) to a set Y with the indiscrete topology. Then f is continuous.

c.) The set of rationals is dense in the usual topology on the real line.

5.) Do the following

a.) Consider the real line with the usual topology. Find the set of limit points and the closure for each of the following sets.

(i) (3, 4) (ii) (3, 4) {7}

(iii) {rationals} (iv)

b.) Consider the set of integers with the cofinite topology. Find the set of limit points and closure for:

(i) {2, 3, . . . , 100}, (ii) {positive integers}

1.) Define the following terms.

a.) Topology b.) Open Set c.) Closed Set

d.) Closure of a Set e.) Dense Set f.) Limit Point

g.) T1 - T2 continuous h.) Interior of a Set i.) Boundary of a Set

j.) Exterior of a Set k.) Base for the Topology, T

2.) Determine if the following are topologies for the given sets. Prove your answer.

a.) X = R, T = {U R {2, 4} U}

b.) X = R, T = {U R U = R or U is closed (in the usual sense) and bounded}

c.) X = R2, A subset of R is said to be radially open if it contains an open line segment in each direction about each of its points. T = {radially open sets}. If this is a topology, is it finer, coarser or neither than the usual topology for R2?

d.) X = R, T = {(a, b) a, b R}

e.) X is an infinite set, T = {U X U = X or U is finite}

3.) Prove the following.

a.) Between any two real numbers is an irrational number.

b.) Definition - Let X be a set and, for each x X, let Sx = {U(x)} be a nonempty family of subsets of X associated with x, such that

(i) x U(x) for each U(x) Sx.

(ii) If V U(x) for some U(x) Sx, then V Sx.

(iii) If U, V Sx, then U V Sx.

(iv) If U Sx, then there exists V Sx such that if y V then U Sy.

Then Sx is called a system of neighborhoods for x. Theorem - Let X be a set and for each x X, let Sx be system of neighborhoods for x. Let T = {U U = or U Sx for some x X}. Then T is a topology for X.

c.) Let (X, T) be a topological space. If A B X then A B. Give an example to show that the reverse inclusion is not true.

d.) Let X be an infinite set with the cofinite topology. Which sets, if any, are dense?

e.) Prove or disprove : On the real line, the cofinite topology is coarser than the usual topology.

4.) Let f: (R, H) (R, U) be the identity map, f(x) = x. Is f an injection? surjection? Is f continuous?

1.) Define the following.

a.) Homeomorphism b.) Disconnected c.) Fixed Point Property

2.) Do the following.

a.) Show that the set B = {[n, n + 1] n an integer} {{n} n an integer} is a base for a topology on R. Is B a base for the half-open interval topology?

b.) Show that a topological space is disconnected if and only if there exists a nonempty proper set that is both open and closed.

c.) Find a base for the discrete topology on R.

d.) Let f be a continuous function from (X, T) onto (Y, S). If X is connected then Y is connected.

e.) Show that any topological space is homeomorphic to itself.

1.) Let f be a function from (X, T) to (Y, S), where BX is a base for T and BY is a base for S. Definition - We say that f is continuous at x X if and only if for every open set V in Y such that f(x) V, there exists an open set U in X such that x U and f(U) V. Prove that f is continuous if and only if f is continuous at each point in X.

2.) Let f be a function from (X, T) to (Y, S), where BX is a base for T and BY is a base for S. Show that f is continuous at x if and only if for each basic open set, U, containing f(x),that there exists a basic open set, V, containing x such that f(V) U.

4.) Let X = {a, b, c}, T = {, X, {a}, {b}, {a, b}}, Y = {x, y}, S = {Y, , {x}}. Define f from X to Y by f(a) = f(b) = x and f(c) = y. Is f continuous? Is f open? Is f a homeomorphism?

1.) Define the following.

a.) Base for the topology, T b.) Connected

c.) Product Space d.) Subspace topology

e.) Homeomorphism f.) Open function

g.) Disconnected h.) Fixed Point Property

2.) Prove the following.

a.) Let B = {(x - 1/n, x + 1/n) x R, n a positive integer} is a base for the usual topology on R.

b.) Let X be a set. Find a base for the indiscrete topology on X.

c.) Let A be a fixed subset of X, where X has a topology, T. Prove that the collection TA = {U A U T} is a topology for A.

d.) Prove that the topology on the integers induced by the half-open interval topology is the discrete topology.

e.) If f is a homeomorphism of (X, T) onto (Y, R) and g is a homeomorphism of (Y, R) onto (Z, S), then gf is a homeomorphism.

3.) Do the following.

a.) Suppose f is a homeomorphism of (X, T) onto (Y, R) and g is a 1-1, continuous function of (Y, R) onto (Z, S). If gf is a homeomorphism, does it follow that g is a homeomorphism?

b.) Let f be an open function from (X, T) onto (Y, S). If X is connected does it follow that Y is connected? If Y is connected does it follow that X is connected?

1.) Define the following terms.

a.) Topological space b.) Regular c.) Closed set

d.) A function from a topological space (X, T) to a topological space (Y, S) is continuous if

e.) A function from a topological space (X, T) to a topological space (Y, S) is a homeomorphism if

f.) Disconnected g.) Compact h.) Metric space

i.) Topological property j

j.) A sequence in a topological space converges to x if

k.) Hausdorff l.) T4

2.) Prove the following.

a.) The real line with the usual topology is Hausdorff.

b.) The real line with the usual topology is not compact.

4.) Prove or disprove. Let X = R and let A be a fixed subset of R. Let T = {U R U = or A U}. Then T is a topology for R.

1.) Define the following terms.

a.) Topological space b.) Open Set

c.) Closed Set d.) Closure of a Set

e.) Dense Set f.) Limit Point

g.) T1 - T2 continuous h.) Interior of a Set

i.) Boundary of a Set j.) Exterior of a Set

k.) Base for the Topology, T l.) Connected

m.) Product Space n.) Subspace topology

o.) Homeomorphism p.) Open function

q.) Disconnected r.) Fixed Point Property

s.) Surjection t.) One-to-one function

u.) Topological property v.) T is finer than S

w.) Cut point x.) L.U.B. of A R

y.) G.L.B. of A R z.) Fixed point of f

aa.) Completeness Property bb.) Compact

cc.) Hausdorff dd.) Regular space

ee.) Normal space ff.) T4, T3, T2, T1, T0

gg.) bounded set hh.) bounded function

ii.) uniformly continuous jj.) Metric

kk.) nested sequence of sets ll.) Metric space

mm.) finite intersection property nn.) Metric topology

oo.) metrizable pp.) sequence

qq.) converges rr.) Cauchy sequence

ss.) complete

2.) Prove the following.

a.) Every metric space is Hausdorff.

b.) Not every topological space is metrizable.

c.) Let d((x1, y1), (x2, y2)) = max{x1 - x2, y1 - y2}. Show that d is a metric for R2.

d.) Let f be a function from the metric space (X, d) to the metric space (Y, p). Then f is continuous if and only if {f(xn)} converges to f(x) whenever {xn} converges to x.

e.) Limits are unique in Hausdorff spaces.

3.) Is the half-open interval topology metrizable?

1.) Let A and B be two fixed non-empty subsets of a set X. Let T be the collection of all subsets of X that are empty, contain A or contain B. Is T a topology for X? If not, would requiring A and B to be disjoint make a difference? If not, would requiring A and B to have nonempty intersection make a difference?

5.) Let {xn} be a sequence in [0, 1]. Prove that there is a convergent subsequence of {xn}. Hint: Bolzano-Weierstrass