Download MAT 371 Advanced Calculus Introductory topology references

MAT 371
Advanced Calculus
Introductory topology references
The definitions and theorems from elementary topology stated here are meant to
give a higher level unifying view of items covered in the MAT 371 textbook, and
to invite students to study more topology in MAT 410.
Definition. A topological space is a set X together with a collection T of subsets of X, called open
n
T
sets, that satisfy: (i) ∅, X ∈ T , (ii) every finite intersection
Uk ∈ T of open sets Uk ∈ T is open,
k=1
and (iii) every union
S
Uα ∈ T of open sets Uα ∈ T is open.
α∈Λ
In a metric spaces (such as IR and IRn with d(p, q) = |q − p| and d(p, q) = kp − qk) a set U ⊂ X
is open if for every p ∈ U there exists ε > 0 such that the open ball Bp (ε) = {q ∈ X: d(p, q) < ε}
is contained in U . (In X = IR, the open balls are the intervals Bx (ε) = (x − ε, x + ε).)
It is common to demand that a topological space X is Hausdorff which means that if for every
p 6= q ∈ X there exist disjoint open sets U, V ⊆ X such that ∈ U and q ∈ V .
Definition. A subset F ⊆ X of a topological space is closed if its complement X \ F is open.
Theorem. A subset F ⊆ X is closed iff it contains all its limit points (accumulation points).
Note the switch of definition and theorem (compared to the MAT 371 textbook).
Definition. A function f : X 7→ Y between topological spaces is continuous
iff for every open set U ⊆ Y the preimage f −1 (U ) ⊆ X is open.
Exercise: show that in metric spaces this definition agrees with the ε-δ-definition.
Definition. A subset K ⊆ X is compact if every open open cover of K has a finite subcover.
Lemma (just the contrapositive – but it takes time to get used to it).
A subset K is compact iff for every collection of closed subsets Fα ⊆ K, α ∈ Λ, for which the
intersection
N
S
k=1
Fαk of any finite subcollection is nonempty, the intersection
S
Fα is nonempty.
α∈Λ
Theorem. Every closed subset of a compact set is compact.
Theorem. Every compact subset of a Hausdorff space is closed.
A subset K ⊆ IRn is compact iff it is closed and bounded. (Heine Borel theorem).
Theorem. If f : X 7→ Y is continuous and K ⊆ X is compact then f (K) ⊆ Y is compact.
If f : X 7→ IR is continuous and K ⊆ X is compact
then there exist p, q ∈ X such that for all x ∈ X, f (p) ≤ f (x) ≤ f (q).
If X, Y are metric spaces, X is compact and f : X 7→ Y is continuous,
then f is uniformly continuous.
Theorem. If f : K 7→ Y is a continuous bijection, K ⊆ X is compact, and Y is Hausdorff, then the
inverse f −1 is continuous.
Theorem. A subset K of a metric space is compact iff it is sequentially compact, i.e., if every infinite
sequence in K has a converging subsequence (with limit in K). (Compare Bolzano Weierstrass.)
All rights reserved. M. Kawski.
http: // math. asu. edu/ ~ kawski
Original version February 17, 2012