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