위상수학 요약정리 Lecture Notes on Topology
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학부 수학과의 위상수학 1/2 에 관한 정의와 정리에 관한 증명입니다.
목차
1 Topological Spaces 5
2 Metric Spaces 11
3 Continuity and Homeomorphism 13
4 Countability 17
5 Connectedness 19
6 Compactness 37
7 Separation Properties 51
8 New space form old Part II 61
본문내용
Chapter 1
Topological Spaces
1.1 Fundamental concepts
De nition 1.1. A topological space is a pair (X; T ) consisting of a set X
and a family T of subsets of X satisfying the following conditions:
(T1) ∅ 2 T and X 2 T
(T2) T is closed under arbitrary union
(T3) T is closed under nite intersection.
The Set X is called a topological space and the subsets of X belongs to T
are called open in the space X.
De nition 1.2. Given a topological space (X; T ) with x 2 X, then N X
is said to be a neighborhood of x if and only if there exist an open set G with
x 2 G N.
Note. It follows then that a set U X is open if and only if for every x 2 U,
9 a neighborhood Nx of x contained in U.
De nition 1.3. Let (X; T ) be a topological space. A family B T is called
a base for (X; T ) if and only if every nonempty open subset of X can be
represented as a union of a subfamily of B.
<중 략>
Theorem 7.7. A T1-space X is regular if and only if 8a 2 X and closed
subset C not containing a, 9 open sets U and V such that
a 2 U; C V and U \ V = ∅:
Proof. ())
Let a 2 X and C is closed set such that a =2 C. Then Cc is open in X
and a 2 Cc. By Theorem 7.6, 9 an open set W such that
a 2 W and W Cc:
Repeating Theorem 7.6, 9 an open set U such that
a 2 U and U W:
Put V = (W)c then
a 2 U U W W Cc:
Thus C (W)c and
U \ V = U \ (W
c
) W \ (W
c
) = ∅:
(()
It is clear by the de nition of regularity.
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