an open ball of radius r centered at a, in a metric space
Let X be a metric space and let a be an element of x for r in R, define Br(a): {x in X: d(a,x)<r></r>
<p>All x in X are within r of a</p>
</r>
an open set, in a metric space
a subset u of a metric space X is _______ if u is the union of open balls
diameter of a set S, in a metric space
Let X be a metric space and let S b contained in X.
- If S =ø then diam(S)=0
- Otherwise diam(S)= sup{d(x,y): x,y are in S} IF sup(S) exists
the convergence of a sequence (an) to a limit x, in a metric space
The sequence (an) in a metric space (X,d) _______ to ________ ________ x in X if
for all N in the Naturals st. n>N, an is in Bepsilon(x)
d(x,an)<epsilon></epsilon>
<p>A tail of the sequence converges to x.</p>
</epsilon>
the Nth tail of a sequence (an), in a metric space
Let (an) be a sequence in (X,d). Let N be an element of the Natural Numbers. The ____ _______ ____ __ ___________ is the subsequence (an)n greater than or equal to N
aN, aN+1, aN+2,…
Theorem 3.4.10
Let S be a subset of R (S is contained in R) and supS=L. There exists an Sn in S s.t. Sn→L
Theorem 3.4.11 Let an and bn be a sequences in R. Let an→a and bn→b. Then, (four things)
- (an + or - bn)→ a + or - b
- (an•bn)→ ab
- (kan)→ ka
- (an÷bn)→ a÷b where b does not equal 0
Let X be a metric space and let S be contained in X
There exists a sequence Sn contained in S whose elements do not include {x} that converges to x
limit point of a set S in a metric space X
A closed set in a metric space
A set C contained in X is _______ if C contains its limit points
Theorem 3.6.3 Let X be a metric space. Let C eb contained in X TFAE:
Can be used to prove that a closed ball is closed
TFAE:
- C is closed
- If (an) is a sequence in C converging to x, then x is an element of C. For all an→x, then x is an element of c
Closed ball of radius r centered at a in a metric space X
Let X be a metric space and let a be an element of X for r in R we define Cr(a)= {x in X: d(a,x) is less than or equal to r}
Boundedness in a metric space
S is bounded in a metric space X if diam(S) exists
Theorem 3.1.12 about S bounded in X TFAE
Let S be contained in X. TFAE:
- diamS exists (S is bounded)
- There exists an a in X and there exists an r>0 Br(a) s.t. S is contained in Br(a)
- For all elements X there exists a Br(a) r>0 s.t S is contained in Br(a)
- for all s in S, d(a,s)<r></r>
Hausdorff Property
for any x, y ∈ X with x =/= y there exist open sets U containing x and V containing y such that U intersect V = ∅.
the intersection of the distance between x and y is empty. The intersection of two disjoint open sets is empty
discrete
If every {x} is open for all x in X, then X is ________
Every point is isolated
Isolated
X is a metric space. x is in X. If {x} is open, then x is isolated. You must be able to fit an open interval around the point
Theorem 3.4.3
Let an be a sequence in R converging to x. Then |an|→|x|.
- There exists a k in R s.t an is greater than or equal to k, then x is greater than or equal to k and |an|→|x|
- If an is less than or equal to k, then x is less than or equal to k
Theorem 3.4.6/7
Let an and bn be sequences in R s.t an→a and bn→b
- If an is less than or equal to bn for all n, then a is less than b
- If an is greater than or equal to bn for all n, then a is greater than or equal to b
Squeeze Theorem
Let an→x and bn→x in R. Sps there exists cn s.t. an is less than or equal to cn is less than or equal to bn, then cn→x
Monotone Sequence Theorem
If an is in R and
- an is bounded
- an is monotone
an is convergent
Every monotone sequence is convergent
Every convergent sequence in R has a _______ _______
bounded subsequence
There is a sequence of distinct points (non-repeating values) in S that converges to x
limit point of a set S in a metric space X
Let X be a metric space and S be contained in X.
For all r>0, Br(x) contains infinitely many points of S
limit point of a set S in a metric space X
Let X be a metric space and S be contained in X.
For all r>0, Br(x) contains one point in S{x}
limit point of a set S in a metric space X
