Thm 27.1: Let X be a simply ____ set having the ______ property. In the order topology, each closed interval in X is ________.

ordered; least upper bound; compact

Corollary 27.2: Every ___ interval in R is cpt

closed

Thm 27.3: A subspace A of R^n is cpt <=> it is ___ and ___ in the ____ and ____ metrics.

closed; bounded; euclidean "d"; square "p"

Thm 27.4: Extreme value thm: Let f: X ->Y be _____, where Y is an ___ set in the ____ topology. If X is ____, then there exists points "c" and "d" in X s.t. f(c) <=f(x)<=f(d) for every x in X

continuous; ordered; order; compact

Defn: distance from A: Let (X, d) be a metric space; let "A" be a nonempty subset of "X." For each x in X, we define the distance from x to A by the equation

d(x, A)=inf{d(x, a)| a in A}.

Lemma 27.5 (The Lebesgue number lemma). Let A be an open covering of the metric space (X, d). If X is ____, there exists a delta>0 s.t. for each ______ of X having _______ less than delta, there exists an element of A containing it.

compact; subset; diameter

(delta is called the Lebesgue number for the covering A).

(delta is called the Lebesgue number for the covering A).

Defn: A function f from the metric space (X, dx) to the metric space (Y, dy) is said to be unifromly continuous if given epsilon>0, there is a delta>0 s.t. for every pair of points x0, x1 of X,

dx(x0, x1)<delta -> dy(f(x0), f(x1))<epsilon

Thm 27.6 (Uniform continuity theorem). Let f: X ->Y be a ______ map of the _____ metric space (X, dx) to the metric space (Y, dy). Then f is ________ continuous.

continuous; compact; uniformly

Defn: If X is a space, a point x of X is said to be an "isolated point" of X if the one-point set {x} is _______ in X.

open

Thm 27.7 Let X be a nonempty ______ Hausdorff space. If X has no isolated points, then X is ________.

compact; uncountable

Corollary 27.8 Every closed interval in R is ______.

uncountable

Defn: A space X is said to be "limit point compact" if every infinite subset of X has a ______ pt.

limit

Thm 28.1 Compactness implies limit point compactness, but not _______.

conversely

Defn: Let X be a topo. space. If (x_n) is a sequence of pts of X, and if n1<n2<...<ni<... is an increasing sequence of positive integers, then the sequence (y_i) defined by setting yi=x_ni is called a ________ of the sequence (x_n).

subsequence

The space X is said to be _______ cpt if every sequence of points of X has a ________ subsequence.

sequentially; convergent

Thm 28.2 Let X be a metrizable space. Then the following are equivalent:

(1) X is _____

(2) X is _____ _____ cpt

(3) X is _________ cpt

(1) X is _____

(2) X is _____ _____ cpt

(3) X is _________ cpt

compact; limit point; sequentially

A space X is said to have a ______ ______ at X if there is a ______ collection B of nbhds of x s.t. each nbhd of x contains at least one of the elts of B.

countable basis; countable

A space that has a countable basis at each of its points is said to satisfy the _____ ________ axiom, or to be _______ _______.

first countability; first-countable

Thm 30.1 (a) Let X be a topo. space. (a) Let A be a subset of X. If there is a sequence of points of A converging to x, then x in A closure; the converse holds if X is ____ _______.

first-countable

Thm 30.1 (b) Let f: X->Y. If f is continuous, then for every________ sequence xn->x in X, the sequence f(xn) converges to f(x). THe converse holds if X is ______ ________.

convergent; first-countable

Defn: If a space X has a countable basis for its topology, then X is said to be _____ ______.

second-countable.

The _____ axiom implies the _____ axiom.

second; first

Thm 30.2a: A subspace of a first-countable space is _____ ______.

first-countable

Thm 30.2b: A countable product of first-countable spaces is ______ ______.

first-countable.

Thm 30.2c: A countable product of second-countable spaces is ______ _______.

second-countable.

A subset A of a space X is said to be "dense" in X if A closure equals ____.

X

Thm 30.3a: Suppose that X has a countable basis. Then, every open covering of X contains a _____ subcollection covering X.

countable

Thm 30.3b: Suppose that X has a countable basis. Then, there exists a _____ subset of X that is _____ in X.

countable; dense

A space for which every open covering contains a countable subcovering is called a _____ space.

Lindelof

A space having a countable dense subset is often said to be

separable.

Lindelof and separable spaces possess the ____ ________ axiom when the space is ______.

second countability; metrizable

The lower limit topo. on R satisfies all the countability axioms but the _______.

second

The real line R has a ______ basis: the collection of all open intvls (a, b) with _______ end points.

countable; rational

In the uniform topology, R^omega satisfies the first countability axiom (being ______). However it does not satisfy the ______.

metrizable; second

The product of two Lindelof spaces need not be ______.

Lindelof;

R_l x R_l is not Lindelof

R_l x R_l is not Lindelof

A _____ of a Lindelof space need not be _____.

subspace; Lindelof

The ordered square is compact; therefore it is Lindelof; However, the subspace A=I x (0, 1) is not Lindelof

The ordered square is compact; therefore it is Lindelof; However, the subspace A=I x (0, 1) is not Lindelof

Defn: regular space: Suppose that one-point sets are ______ in X. Then X is said to be "regular" if for each pair consisting of a point x and a _______ set B disjoint from x, there exist _______, _________ sets containing x and B respectively.

closed; closed; disjoint, open

Defn: normal space: The space X is said to be "normal" if for each pair A, B of disjoint _____ sets of X, there exist disjoint _____ sets, containing A and B respectively

closed; open

A regular space is _______.

Hausdorff

A normal space is _______.

regular

Regularity axiom stronger than the ______ axiom.

Hausdorff

Normality stronger than _______.

regularity

Lemma 31.1a: Let X be a topological space. Let one-point sets in X be closed. X is ______ <=> given a point x of X and a nbhd U of x, there is a nbhd V of x s.t. _______ subset of U

regular; closure(V)

Lemma 31.1b: X is _______ <=>given a closed set A and an _____ set U containing A, there is an open set V containing __ s.t. closure(V) subset of ____

normal; open; A; U

Thm 31.2a) A subspace of a Hausdorff space is ________; a product of ______ spaces is Hausdorff

Hausdorff; Hausdorff

Thm 31.2b) A subspace of a ______ space is ______; a product of ______ spaces is ______.

regular; regular; regular; regular

Ex: The space R_k is Hausdorff but not _______.

regular

Ex: THe space R_l is _______.

normal

Ex: The Sorgenfrey plane R_l x R_l is ___ _______.

not normal

If X is regular, every pair of points of X have nbhds whose closures are _________.

disjoint

Every order topology is ________.

regular

Thm 32.1: Every regular space with a _______ basis is ______.

countable; normal

Thm 32.2: Every _________ space is normal

metrizable

Thm 32.3: Every _______ Hausdorff space is ______.

compact; normal

Thm 32.4: Every _______ ______ set X is normal in the _____ topology

well-ordered; order

Ex: If J is uncountable, the product space R^J is not ______.

normal

A closed subspace of a ______ space is _______.

normal; normal

Every regular ________ space is _______.

Lindelof; normal

A space is said to be "completely normal" if every subspace of X is ______.

normal

X is completely normal <=> for every pair A, B of separated sets in X, there exist _____ _____ sets containing them

disjoint; open

A ________ normal space having more than one point is _________.

connected; uncountable

Any countable space is ________.

Lindelof

A connected ________ space having more than one point is _______.

regular; uncountable

Thm 33.1: Urysohn lemma: Let X be a ______ space; let A and B be disjoint ______ subsets of X. Let [a, b] be a closed interval in the ______ line. Then there exists a continuous map f: X->[a, b] s.t. f(x)=a for every x in A and f(x)=b for every x in B

normal; closed; real

Defn: If A and B are two subsets of the topo. space X, and if there is a _________ function f: X-> [0, 1] s.t. f(A)={0} and f(B)={1}, we say that A and B can be "________ by a continuous function"

continuous; separated

Defn: A space X is "________ regular" if one-point sets are closed in X and if for each point x0 and each _______ set A not containing x0, there is a continuous function f: X->[0, 1] s.t. f(x0)=1 and f(A)={0}

completely; closed

Thm 33.2: A subspace of a ________ regular space is _______ regular. A product of ______ regular spaces is _____ _________.

completely; completely; completely; completely regular

Thm 34.1 (Urysohn metrization thm): Every regular space X with a ______ basis is _________.

countable; metrizable

Give a direct proof of the Urysohn lemma for a metric space (X, d) by setting f(x)=

f(x)=d(x, A)/( d(x,A)+d(x, B) )

Let X be a compact, _______ space. Show that X is metrizable <=> X has a ______ basis.

Hausdorff; countable

Thm 34.2 (Imbedding Thm). Let X be a space in which one-point sets are closed. Suppose that "{f_\alpha}_\alpha \in J" is an indexed family of continuous functions "f_\alpha: X \to R" satisfying the requirement that for each point x0 of X and each nbhd U of x0, there is an index "alpha" s.t. "f_\alpha" is _____ at x0 and ______ outside U. Then the function F: X -> R^J defined by "F(x)=(f_\alpha(x))_\alpha \in J" is an ________ of X in R^J. If f_\alpha maps X into [0, 1] for each \alpha, then F imbeds X in [0, 1]^J

positive; vanishes; imbedding

Thm 34.3: A space X is completely _______ <=> it is ________ to a subspace of [0, 1]^J for some J

regular; homeomorphic

Thm 35.1a (Tietze extension thm). Let X be a ______ space; let A be a _____ subspace of X. Then, any continuous map of A into the closed interval [a, b] of R ma be extended to a ______ map of all of X into [a, b].

normal; closed; continuous

Thm 35.1b (Tietze extension thm). Any continuous map of A into R may be extended to a _____ map of all of X into R

continuous