The initial Topology theorems from scratch.

1. On Limits and Convergence
vinicius.costa
16 June 2019
1 Introduction
In these definitions, for the sake of clarity, I tried to make sure that every
definition uses only previously defined terms.
In general, however, there’s some divergence on how neighborhoods and open
sets can be defined. Most professional mathematicians have acquired both an
intuitive and practical understanding of what these are, and as a result the
effort of remembering and sticking to a formal definition starts not to matter
anymore.
When doing research, they know the rules of dealing with these concepts by
heart, and that is basically all that is needed to advance research.
The problem is when a novel student is seeing these terms for the first time,
and is confronted with divergent definitions which use each other in a circular
fashion.
Definition 1.1 Let a ∈ Rn
and let δ > 0. The ball of radius δ centered at a is
the set
B(a, δ) = {x ∈ Rn
: ||x − a|| < δ}
Note the inequality is strict: we’re not including the outer surface of the ball as
part of the ball.
Definition 1.2 A neighborhood of a point a is a set U that contains at least
one ball centered in a.
Remark: for the sake of practicality, we defined neighborhood in such a way
that to show U is a neighborhood of a we need only show there’s at least one
a−centered ball in U . As the reader can predict, every neighborhood of a we
can come up with will actually have infinitely many such balls: just keep taking
smaller radii.
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2. Definition 1.3 An open set is a set S with the property that for every a ∈ S,
S contains a ball centered in a.
Definition 1.4 An open neighborhood of a is a neighborhood of a point a that
is also an open set.
The diagram below shows the dependency of the concepts presented so far.
Open Neighborhood
NeighborhoodOpen Set
Ball
Now, we move on to define convergence of a sequence:
Definition 1.5 (Definition of a Limit, by Shifrin) We say that the sequence
{xk} converges to a (denoted xk → a or lim
k→∞
xk = a) if for all > 0, there is
K ∈ N such that
||xk − a|| < whenever k > K.
(That is, given any ball centered at a, “eventually”—past some K—all the ele-
ments xk of the sequence lies inside.) We say the sequence {xk} is convergent
if it converges to some a.
Theorem 1.1 (Convergence via Auxiliary Sequence) Let {yk} be a se-
quence with lim
k→∞
yk = 0.
If {xk} is a sequence such that for large enough k, ||xk − a|| ≤ yk, i.e.,
∃ N ∈ N such that ∀ k > N : ||xk − a|| ≤ yk,
then lim
k→∞
xk = a.
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3. Proof: Take an arbitrary > 0. Because yk → 0, then:
∃N0 ∈ N such that ∀k > N0 : ||yk − 0|| = ||yk|| < .
Therefore, take N = max (N0, N). We will have:
∀k > N : ||xk − a|| ≤ yk ≤ ||yk|| < .
⇒ lim
k→∞
xk = a
Corollary 1.1 (Convergence via 1
n criterion) Let a ∈ Rn
. If a sequence
{xk} is such that
∀k ∈ N, ||xk − a|| ≤
1
k
,
then the sequence converges to a.
Definition 1.6 A closed set is a set S with the property that every convergent
sequence {xk} with all elements in S converges to an element of S.
Proposition 1.1 The subset S ⊂ Rn
is closed if and only if its complement,
Rn
− S = {x ∈ Rn
: x /∈ S} is open.
Suppose Rn
− S is open and {xk} is a convergent sequence with xk ∈ S and
limit a. Suppose that a /∈ S. Then there is a neighborhood B(a, ) of a wholly
contained in Rn
− S, which means no element of the sequence {xk} lies in that
neighborhood, contradicting the fact that xk → a. Therefore, a ∈ S, as desired.
Suppose S is closed and b /∈ S. We claim that there is a neighborhood of b
lying entirely in Rn
− S. Suppose not. Then for every k ∈ N, the ball B(b, 1/k)
intersects S; that is, we can find a point xk ∈ S with ||xk − b|| < 1/k. Then,
by Corollary 1, {xk} is a sequence of points in S converging to the point b /∈ S,
contradicting the hypothesis that S is closed.
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