nopu@wiki

実数の位相

Def. Open Sets
A set O⊂R is open if for all points a∈O there exists an ε-neighborhood V(a;ε)⊂O

Def. Limit point
A point x is a limit point of a set A if every ε-neighborhood V(x;ε) of x intersects the set A in some point other than x.
They are also often referred to as cluster points or accumulation points

Rem. A limit point of A may not belong to A
As an example, consider the endpoint of an open interval.

Th. A limit point is the limit of a sequence.
A point x is a limit point of a set A if and only if x=lim an
for some sequence (an) contained in A satisfying an≠x for all n∈N

Def. Isolated Point
A point a∈A is an isolated point of A if it is NOT a limit point of A.

Def. Closed Set
A set F⊂R is closed if it contains its limit points.

Th.
A set F⊂R is closed if and only if every cauchy sequence contained
in F has a limit that is also an element of F.

Def. Closure
Given a set A⊂R, let L be the set of all limit points of A.
The closure of A is defined to be A∪L.

Th. 
For any A⊂R, the closure A- is a closed set and is the smallest closed set containing A.

Def. Compact Sets
A set K⊂R is compact if every sequence in K has a subsequence
that converges to a limit that is also in K

Th. Heine-Borel Theorem
A set K⊂R is compact if and only if it is closed and bounded.

Ex. 
The Cantor set is compact.

Def. Open Covers

Ex. 
Consider the open interval (0,1).
For each point x∈(0,1), let Ox be the open interval (x/2, 1).
Taken together, the infinite collection {(Ox) : x∈(0,1)} forms
an open cover for the open interval (0,1).
Notice that it is impossible to find a finite subcover.

Th. a finite subcover
Let K⊂R. K is compact if and only if any open cover for K has a finite subcover.

Def. Perfect Sets
A set P⊂R is perfect if it is closed and contains NO isolated points.

Ex. 
The Cantor set is perfect.

Th. 
A nonempty perfect set is uncountable.

Def. Connected Sets
Two nonempty sets A,B⊂R are separated if A-∩B and A∩B- are both empty.
A set E⊂R is disconnected if it can be written as the union of two nonempty separated sets.
A set that is NOT disconnected is called a connected set.

Ex. The set of rational numbers is disconnected.

Th. 
A set E⊂R is connected if and only if whenever a<c<b with a,b∈E, it follows that c∈E as well.

Def. Nowhere-Dense Sets
A set E is nowhere-dense if E- contains no nonempty open intervals.

Th. Baire's Category
The set of real number R cannot be written as the countable union of nowhere-dense sets.

Rem. three perspectives of the size of R
uncountable, non-zero measure, second category