Advertisement Upgrade to remove ads

math

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).

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

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

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

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

Please allow access to your computer’s microphone to use Voice Recording.

Having trouble? Click here for help.

We can’t access your microphone!

Click the icon above to update your browser permissions above and try again

Example:

Reload the page to try again!

Reload

Press Cmd-0 to reset your zoom

Press Ctrl-0 to reset your zoom

It looks like your browser might be zoomed in or out. Your browser needs to be zoomed to a normal size to record audio.

Please upgrade Flash or install Chrome
to use Voice Recording.

For more help, see our troubleshooting page.

Your microphone is muted

For help fixing this issue, see this FAQ.

Star this term

You can study starred terms together

NEW! Voice Recording

Create Set