위상수학

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위상수학에 대한 보고서 자료입니다.

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Let ,
then
is continuous
since
※ : continuous (i.e. )
Define
then
thus : continuous
Define
then : continuous
such that
Define , i.e.
then : continuous
Theorem 7.11
: metric space,
then
is continuous
pf.
Corollary 7.11.1
: metric space
pf. Since is continuous, : closed in
: closed
Let
such that
and
Theorem 7.12
: metric space
then : continuous
pf.
Chapter 8. Normal Space
Definition
space
: normal orspace
closed set in
open such that
※
Example
space
space
※ : normal, : normal : normal (John's Theorem)
Theorem 8.1
: normal closed ,open
open such that
pf. Let : closed, : open,
: closed, : disjoint
by definition of normal
: open, such that
Let : closed in , (open)
by hypothesis, open such that
Theorem 8.2
: continuous, closed, bijective (homeomorphism)
: normal
pf. Let : disjoint closed in
: disjoint closed in
open such that
since : homeomorphism
: open such that
Theorem 8.3
: metric space : normal
pf. Let : metric on
: closed in ,
Let
then
Theorem 8.4 (Urysohn Lemma)
space
: normaldisjoint nonempty closed in
continuous such that
pf. Let : closed in ,
let continuous,
then
: open in
Let
Example
then
if
: even
↑
Dyadic number
Since is normal
let : closed,
Construct : open, such that
(a)
(b)
(1)
Let ( : open) and (:closed)
: open such that
(2) Assume such that satisfies (a), (b)
if
such that
, We can construct
Define by inf
then and : continuous
(1) Let
take such that
let
if then
: continuous
(2) Let
such that
(3) Let
such that
※ For : closed set, : continuous
assume continuous such that
if : closed,
and be map as
by hypothesis, continuous such that
let , ,
: open
Theorem 8.5 (Tietze's theorem)
: normalclosed set continuous
continuous such that
Definition
: topological space
: continuous
Supp : support of Supp
Theorem 8.6
space
: normal : closed,open such that
such that ①
②
③ Supp
pf. Let : closed sets
then
by normality, such that
①
②
③
Let : closed sets
by normality, such that
①
②
③
if
Supp
: disjoint closed set
: closed, : open
by hypothesis, such that ①
②
③ Supp
Let : open sets
Definition
: topological space
: locally finite
such that : finite
Definition
: topological space
: partition of unity
①
②Supp : locally finite
: partition of unity ordinoted by
③ Supp
Definition
: topological space
: cover of
: cover of and : open : open cover
Theorem 8.7
: normal
: open cover of
: locally finite
: partition of unity such that Supp
pf. : open cover of
then closed such that
: cover of
by normality, such that
①
②
③ Supp
since : locally finite
let
then : continuous,
define : continuous
SuppSupp
finally,
Definition
: topological space
: Completely regular
closed such that
①
②
③
space, completely regular : Tychonoff space
※ : completely regular,
: completely regular
pf. : closed in
: closed in
such that
①
②
③
: continuous
①
②
③
※ : normal : normal
: completely regular
: completely regular : regular
pf. Let : open neighborhood of
such that
let : open
: regular

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