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topological space
Definition
: Neighborhood of (nbd of )
such that
: nbd of
Example1.1
, ,
(1)
: not open, : open
: nbd of
(2)
: open
: nbd of
(3)
: open, but : not open
: nbd of
(4)
: open
: nbd of
Example1.2
: set
: Topology (Indiscrete Topology) on
: Topology (Discrete topology) on
Example1.3
: set,
: topology on
sol. (1)
(2) Let , then we have two cases
①
② or
(3) If
Example1.4
: set
: topology (Cofinite topology or Finite complement topology) on
sol. (1)
(2) If
(3) If
◆ : topological space,
: nbd of then we have
Theorem 1.1
(1)
(2)
(3)
(4) such that
pf. (3) Let
: open in , such that
(since : open)
(4) Let
: open such that
let then
Theorem 1.2
: open
pf. : open, such that
Since : open, such that
: open
Definition
: set
satisfies
(1)
(2)
(3)
(4)
then is said to be a metric on
: metric space with metric d
◆
Example1.5
then : metric space (Pythagorian metric)
◆ : taxi cab metric
Example1.7
Let be a set
: injective
Define by
then : metric on
pf. (1)
(2)
(3)
(4)
: injective
Example1.8
then : metric on , and
pf. Let by
then : injective
(1)
(2)
: contradiction
(3)
Example1.9
: metric space
Define by inf
then : metric on
◆ : metric space
inf
◆
pf.
, for some
Definition
sup : diameter
: bounded
◆
: metric
: metric on
: dim sphere
: metric space and bounded
Definition
: metric space,
: neighborhood of
◆ : metric space
: open such that
: induced topology by metric
pf. (1)
(2)
such that
such that
take
(3)
such that
◆ : : usual metric
: real space ( : usual topology)
Example1.10
① : open set
② : not open set
sol. ① Take
then open
② Let and
Definition
: topological space
: closed set :open set
Example1.11
: closet set
sol. Take
then : open set
: closed set
Example1.12
,
: closed set
: open set
pf.
: open set
: closed set
◆ Thus we can have
(1) : closed set
(2) : closed set : closed set
(3) : closed set : closed set
pf. (2)
open ∩ open open
(3) : open
Definition
: topological space,
Define
then : topological space(subspace of )
: subspace topology or relative topology
pf. (1)
(2)
(3)
◆ : closed in A : open in
, : open in
: closed in
: closed in , : closed in
: closed in
◆
: subspace
: open in
: not open in
Definition
: topological space
open , such that
( such that )
oropen , such that
: Kolmogorov space or space
Example1.13
(1)
: space
such that
such that
such that
(2)
: notspace
Definition
: topological space
and such that
: Frechet space or space
Example1.14
and
: not space
◆ space : closed set
pf. Let
such that
: open set
: closed set
Let
: closed set
: open set, : open set
: open set, : open set
,
space
Definition
: topological space
: Hausdorff space or space
such that
◆ space spacespace
Theorem 1.3
: space,
space
pf.,
such that , : open
and
space
Theorem 1.4
space,
: closed
Theorem 1.5
: metric space
space
pf.
take
and
Example1.15
: topological space
: open
: open,
: Discrete topological space
space
pf.
: open in
: open in
: discrete topology

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