Fundamentals Flashcards
Real Numbers Constructed
are rational Cauchy sequences
up to equivalence
{an} ~ {bn} if |ar - br| < e
“corresponding terms”
for all r > some M
(e in Q+ )
What does it mean for real number x to be positive?
it means for all n > N,
the sequence {a_n} representing x > some delta
“bounded away from 0”
What is are the reals axiomatically?
field
ordered ( trichotomy, archimedean)
completeness
or
l.ub.
or
every increasing, bounded sequence converges
what is the lim sup of an ?
for SN = { aN, aN+1, …}
limN -> inf SN
exists b/c S_N is bounded and decreasing
“largest tail”
*every bounded sequence has a lim sup
how do we define convergence in terms of lim sup?
{an} converges
<==>
lim sup = lim inf
continuity in any metric space
close inputs yield close outputs
(in appropriate metric)
What is an Open Ball?
a A set is open if
for every x,
there is an open ball centered at x
entirely contained in A
(open ball center x : {y : d(x, y) < radius})
* open ball is an open set
a topological space (Z, T) is …
a set Z and subsets of Z, labeled T such that
1) Ø and Z in T
2) any union of T is in T
(even infinite)
3) finite intersections in T are in T
Continuity in topological spaces
traditional closeness definition
or
all open sets in range
map to
open sets in the domain
A topological space (Z, T) is Hausdorff
(or T2-space)
if for all x, y in Z
x and y can be seperated
by disjoint open sets
Cardinality of
|Ø| is …
|P(Ø)| = powerset of empy is …
for a set A,
|A| (>, < ?) |P(A)|
|Ø| is 0
|P(Ø)| = 1
“one subset of empty”
|A| < |P(A)| = 2^|A|
an inifinite union of countable sets is
countable!
Axiom of Choice says…
the Cartesian product of non-empty sets
is non-empty
Haudorff Space
topological space
where points can be separated by open balls
Open Set, O
every point can be surrounded by an open ball
entirely contained in O
