Chapter 3 Flashcards
A set O c R is open if…
for all points a in O, there exists an ε-neighborhood Vε(a) c O
for all points a in O, there exists an ε-neighborhood Vε(a) c O
A set O c R is open if…
Theorem-
i. ) The union of an arbitrary collection of open sets is…
ii. ) The intersection of a finite collection of open sets…
i. ) … is open
ii. ) is open.
A point is a limit point of a set A if…
every ε-neighborhood Vε(x) of x intersects the set A at some point other than x.
every ε-neighborhood Vε(x) of x intersects the set A at some point other than x.
A point is a limit point of a set A if…
Theorem- A point x is a limit point of a set iff…
x = lim an for some sequence (an) contained in A satisfying an not equal to x for all n in N.
x = lim an for some sequence (an) contained in A satisfying an not equal to x for all n in N.
Theorem- A point x is a limit point of a set iff…
A point a in A is an isolated point of A if…
it is not a limit a point of A.
it is not a limit a point of A.
A point a in A is an isolated point of A if…
A set F c R is closed if…
F contains all its limit points.
F contains all its limit points.
A set F c R is closed if…
Theorem- A set F c R is closed if and only if…
every Cauchy sequence contained in F has a limit that is also an element of F.
every Cauchy sequence contained in F has a limit that is also an element of F.
Theorem- A set F c R is closed if and only if…
Density of Q in R- for every y in R, …
there exists a sequence of rational numbers that converges to y.
there exists a sequence of rational numbers that converges to y.
Density of Q in R- for every y in R, …
Given a set A c R, let L be the set of all limit points of A.
The closure of A is defined as…
Ā = A U L
Ā = A U L
Given a set A c R, let L be the set of all limit points of A.
The closure of A is defined as…
Theorem- For any A c R, the closure Ā is a closed set and …
the smallest closed set containing A.
the smallest closed set containing A.
Theorem- For any A c R, the closure Ā is a closed set and …
Theorem-
i. ) A set O is open iff…
ii. ) A set F is closed iff…
i. ) the complement Oc is closed.
ii. ) the complement Fc is open.
i. ) the complement Oc is closed.
ii. ) the complement Fc is open.
Theorem-
i. ) A set O is open iff…
ii. ) A set F is closed iff…
A set K c R is compact if…
every sequence in K has a subsequence that converges to a limit that is also in K.
every sequence in K has a subsequence that converges to a limit that is also in K.
A set K c R is compact if…
A set A c R is bounded if…
there exists an M > 0 such that |a| < M for all a in A.
there exists an M > 0 such that |a| < M for all a in A.
A set A c R is bounded if…
Characterization of Compactness in R- A set K c R is compact…
iff it is closed and bounded.
iff it is closed and bounded.
Characterization of Compactness in R- A set K c R is compact…
Nested Compact Set Property-
If … K2 c K1
is a nested sequence of non-empty compact sets, then…
the intersection of Kn is not empty.
the intersection of Kn is not empty.
Nested Compact Set Property-
If … K2 c K1
is a nested sequence of non-empty compact sets, then…
Let A c R. An open conver for A is…
a possibly infinite collection of open sets whose union contains the set A.
a possibly infinite collection of open sets whose union contains the set A.
Let A c R. An open conver for A is…
Given an open cover for A, a finite subcover…
is a finite subcollection of open sets from the original open cover whose union still contains A.
is a finite subcollection of open sets from the original open cover whose union still contains A.
Given an open cover for A, a finite subcover…
Heine-Borel Theorem- Let K be a subset of R. All of the following statements are equivalent in the sense that any one of them implies the two others:
i. ) K is compact.
ii. ) K is closed and bounded.
iii. ) Every open cover for K has a finite subcover.
i. ) K is compact.
ii. ) K is closed and bounded.
iii. ) Every open cover for K has a finite subcover.
Heine-Borel Theorem- Let K be a subset of R. All of the following statements are equivalent in the sense that any one of them implies the two others:
