9. Homeomorphisms(위상동형사상)
Homeomorphisms (위상동형사상)
Definition. topological spaces
bijective일 때,
is a homeomorphism
⇔ is continuous and is also continuous.
Remark : ① Every homeomorphism (and its inverse) is an open mapping : is also open for any open in .
That is, (by def.) is open, since ⊇ ↦ , and is continuous
② Similarly, every homeomorphism (and its inverse) is a closed mapping : is closed for any closed set in . That is,
is closed, since is continuous.
Definition: injective, continuous;
-> is bijective. If → happens to be homeomophism, then is called a topological imbedding or just imbedding (embedding) of onto
e.g.
(1) cos
sin
ⓐ is 1-1 and continuous ⊆ -> is of the form
= ∪ ∞∪ ∞ open
ⓑ equipped with the induced topology form ; is also continuous by the reason as above.
ⓒ But, → is not a homeomorphism with respect to the induced topology;
∩
∪ ∞∩ ∞
= as above "three connected components"
are connected component. In fact, when ,
: not open in the induced topology.
ⓓ Next, we put a different topology on ;
a sub-basis for a topology as above
~> ∞ ∪
∪ ∞ not open ;
(2) ∀
is continuous, 1-1, but not homeomorphism, since is not continuous!
(3)
cos sin
1-1, continuous, onto
But is not continuous ; e.g.
[///) )
Constructing continuous functions Basic facts about continuous functions
topological spaces ① is continuous ↦ = constant
② ⊆ -> ↪ is continuous inclusion subspace
③
④ &
⊇
->
is continuous
⑤
⑥
⑦ is continuous of can be written as union of open sets ∝
s.t. ∝ is continuous for each ∝.
Proof: ① ∀ open of in
⇒ is open in ∴ is continuous
② open in ⇒ ∩ ⇒
def is open in ∴ ↪ is continuous.
③ ∀ open in
~> is open in ->
conti ∘ is open in
④
↪
→
~>
∘ & and are continuous.
~>
is continuous
⑤ open in ⇒ ∩ open in ⇒ ∩
∩
∩
: open
∵ is continuous
⑥ ∼
∘ & are continuous ~> ∼
is continuous on
⑦ ∀ open in
║
∝ ∩∝
∝
∝ ： open in
q.e.d.
Theorem (Pasting lemma):
∪ ⊆ closed (resp. open) →
→
∈∩ or
Then,
∈ ∈ are continuous well-defined over ∪Proof : ⊂ closed subset of
⇔ ∈ ⇔ ∈ or ∈
∈ ∈⇔ ∈ or , since is continuous, from to .
Thus, is closed in ⇒
is closed in Similarly, since is continuous from to ,
is closed in ⇒ is closed in ⇒ ∪ is closed in
q.e.d.
As an application, let → continuous. Here is a totally ordered set with the order topology.
Let min be a function given by
≥ ≤ .
(i) well-defined, since is defined to be where
But the set ∈ ≤ is closed∗, and ∈ ≥ is also closed
is continuous
~> is continuous on by Pasting Lemma.
q.e.d.
In particular, in case of , we have
min
max
. Proof of
:Claim : ∈ is open in (Assume otherwise we're done)
Let ∈ Then . We may assume without loss of generality that there should be some element ∈ s.t.
.
(Otherwise, to )
moreover,
∈
∩⊆ ∀ ∈∩ ⇒
⇒
∴ ∈
Theorem ×
↦
Then, is continuous ⇔ → and → are both continuous.
Proof : ⇒ ×
↓
are projections onto the 1st and 2nd
factors are continuous, and open mapping. Thus, ¡ ¡∘ is continuous
⇐ ∀× basis element for the product × × is open ?
But, × ∩ ⊆
⊇
∈ × ⇔
∈× ⇔ ∈ ∩ ~> × is open.