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MAT 410 Intro to General Topology
Spring 2006
Basic logic example: proofs of compactness using open covers
A common structure of many theorems is an implication of the form
(A ⇒ B) ⇒ (C ⇒ D)
for suitable compound statements (or propositions) A, B, C, and D which generally may
involve various quantifiers as well. On a formal level such implication is logically equivalent
to
(C ∧ (A =⇒ B)) ⇒ D
Exercise: Prove this logical equivalence by rewriting implications as disjunctions and suitably distributing negations over disjunctions or conjunctions.
The general advice is to start the proof with “Suppose C (is true)”, then use along the way
that A implies B to conclude D. Of course, there are many other approaches which may be
more suitable in any particular instance, and which may better accommodate the internal
structure of the propositions A, B, C, and D.
Nice applications are basic proofs involving compactness. Recall, a subset K of a topological
space (X, T ) is compact if every open cover of K contains a finite subcover of K.
Typical basic theorems have the logical structure
“(Suppose . . . something about A and B . . . ) If A is compact, then B is compact.”
This means that
“If every open cover of A contains a finite subcover of A,
then every open cover of B contains a finite subcover of B.”
or, written
à more symbolically
Ã
!!
[
[
∀U ⊆ T , A ⊆
U ⇒ ∃U0 ⊆ U finite s.t. A ⊆
U
Ã
!!
U ∈U Ã
U ∈U0
[
[
=⇒ ∀V ⊆ T , B ⊆
V ⇒ ∃V0 ⊆ V finite s.t. B ⊆
V
V ∈V
V ∈V0
To prove a statement of this form, the default strategy is
(i) Suppose V is an open cover of B.
(ii) Use V to construct a related open cover U of A.
(The term “related” is used here in a very loose sense.)
(iii) Use the hypothesis to conclude that U contains a finite subcover U0 of A.
(iv) Go back from U0 to exhibit a finite subcover V0 ⊆ V of B.
Exercises: Apply this above strategy to prove that
• closed subsets of compact sets are compact,
• finite unions of compact sets are compact,
• finite products of compact sets are compact,
• continuous images of compact sets are compact,
• continuous images of connected sets are connected.
http://math.asu.edu/˜kawski
Draft, April 3, 2006. Matthias Kawski,