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Complete Metric Spaces
Definition: Let (X,d) be a metric space. A sequence
(x_n) is called Cauchy if
A sequence (x_n) is called convergent if there
exists an x in X such that
Definition: A metric space is caled complete if every Cauchy is
convergent.
Definition: A subset D in topological space is dense it the X=D,
the closure of D.
Recall that the closure of a set is the smallest closed set
containing it.
Definition: In a metric space X a point x belongs to the interior
of
a set V if B(x,r) is contained in V for some r>0.
Definition: A set D in a metric space is called nowhere dense if
The closure has non-empty interior.
Theorem(Baire): In a complete metric space the countable
union of nowhere dense sets is again nowhere dense.