aarjaneiro/planetmath huggingface dataset:
order topology

order topology

Let

(

X

,

≤

)

be a

linearly ordered set

. The

order topology

on

X

is defined to be the

topology

𝒯

generated by the

subbasis

consisting of open rays, that is sets of the form

(

x

,

∞

)

=

{

y

∈

X

|

y

>

x

}

(

-

∞

,

x

)

=

{

y

∈

X

|

y

<

x

}

,

for some

x

∈

X

.

This is

equivalent

to saying that

𝒯

is generated by the basis of

open intervals

; that is, the open rays as defined above, together with sets of the form

(

x

,

y

)

=

{

z

∈

X

|

x

<

z

<

y

}

for some

x

,

y

∈

X

.

The

standard topologies

on

ℝ

,

ℚ

and

ℕ

are the same as the order topologies on these sets.

If

Y

is a subset of

X

, then

Y

is a linearly ordered set under the

induced

order from

X

. Therefore,

Y

has an order topology

𝒮

defined by this

ordering

, the

induced order topology

. Moreover,

Y

has a

subspace topology

𝒯

′

which it inherits as a

subspace

of the

topological space

X

. The subspace topology is always finer than the induced order topology, but they are not in general the same.

For example, consider the subset

Y

=

{

-

1

}

∪

{

1

n

∣

n

∈

ℕ

}

⊆

ℚ

. Under the subspace topology, the

singleton set

{

-

1

}

is open in

Y

, but under the order topology on

Y

,
any

open set

containing

-

1

must contain all but finitely many members of the space.

A chain

X

under the order topology is

Hausdorff

: pick any two distinct points

x

,

y

∈

X

; without loss of generality, say

x

<

y

. If there is a

z

such that

x

<

z

<

y

, then

(

-

∞

,

z

)

and

(

z

,

∞

)

are

disjoint

open sets

separating

x

and

y

. If no

z

were between

x

and

y

, then

(

-

∞

,

y

)

and

(

x

,

∞

)

are disjoint open sets separating

x

and

y

.

Title

order topology

Canonical name

OrderTopology

Date of creation

2013-03-22 12:10:34

Last modified on

2013-03-22 12:10:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54B99

Classification

msc 06F30

Synonym

induced order topology

Related topic

OrderedSpace

Related topic

LinearContinuum

Related topic

ProofOfGeneralizedIntermediateValueTheorem

Related topic

ASpaceIsConnectedUnderTheOrderedTopologyIfAndOnlyIfItIsALinearContinuum

Generated on Sat Feb 10 11:17:44 2018 by

LaTeXML
suspension

suspension

1

The unreduced suspension

Given a

topological space

X

,

the

suspension

of

X

,

often denoted by

S

⁢

X

,

is defined to be the

quotient space

X

×

[

0

,

1

]

/

∼

,

where

(

x

,

0

)

∼

(

y

,

0

)

and

(

x

,

1

)

∼

(

y

,

1

)

for any

x

,

y

∈

X

.

Given a

continuous map

f

:

X

→

Y

,

there is a map

S

⁢

f

:

S

⁢

X

→

S

⁢

Y

defined by

S

⁢

f

⁢

(

[

x

,

t

]

)

:=

[

f

⁢

(

x

)

,

t

]

.

This makes

S

into a

functor

from the category of topological spaces into itself.

Note that

S

⁢

X

is

homeomorphic

to the join

X

⋆

S

0

,

where

S

0

is a

discrete space

with two points.

The space

S

⁢

X

is sometimes called the

unreduced

,

unbased

or

free

suspension of

X

,

to distinguish it from the reduced suspension described below.

2

The reduced suspension

If

(

X

,

x

0

)

is a

based topological space

, the

reduced suspension

of

X

,

often denoted

Σ

⁢

X

(or

Σ

x

0

⁢

X

when the

basepoint

needs to be explicit), is defined to be the quotient space

X

×

[

0

,

1

]

/

(

X

×

{

0

}

∪

X

×

{

1

}

∪

{

x

0

}

×

[

0

,

1

]

.

Setting the basepoint of

Σ

⁢

X

to be the

equivalence class

of

(

x

0

,

0

)

,

the reduced suspension is a functor from the

category

of based topological spaces into itself.

An important

property

of this functor is that it is a

left adjoint

to the functor

Ω

taking a (based) space

X

to its

loop space

Ω

⁢

X

. In other words,

Maps

*

(

Σ

⁢

X

,

Y

)

≅

Maps

*

(

X

,

Ω

⁢

Y

)

naturally, where

Maps

*

(

X

,

Y

)

stands for continuous maps which

preserve

basepoints.

The reduced suspension is also known as the

based

suspension.

Title

suspension

Canonical

name

Suspension

Date of creation

2013-03-22 13:25:37

Last modified on

2013-03-22 13:25:37

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

10

Author

antonio (1116)

Entry type

Definition

Classification

msc 54B99

Related topic

Cone

Related topic

LoopSpace

Related topic

Join3

Related topic

SuspensionIsomorphism

Defines

suspension

Defines

reduced suspension

Defines

based suspension

Defines

unreduced suspension

Defines

unbased suspension

Generated on Sat Feb 10 11:17:47 2018 by

LaTeXML
classification of topological properties according to behaviour under mapping

classification of topological properties according to behaviour under mapping

Topological properties may be classified by their behaviour with respect to mappings. The basis of such a

classification

is the following question: Given two

topological spaces

X

and

Y

and a

continuous map

f

:

X

→

Y

, can one infer that one of the spaces has a certain topological property from the fact that the other space has this property?

A trivial case of this question may be disposed of. If

f

is a

homeomorphism

, then the spaces

X

and

Y

cannot be distinguished using only the techniques of

topology

, and hence both spaces will have exactly the same topological properties.

To obtain a non-trivial classification, we must consider more general maps. Since every map may be expressed as the

composition

of an inclusion and a surjection, it is natural to consider the cases where

f

is an inclusion and where it is a surjection.

In the case of an inclusion, we can define the following classifications:

A property of a topological space is called

hereditary

if it is the case that whenever a space has that property, every

subspace

of that space also has the same property.

A property of a topological space is called

weakly hereditary

if it is the case that whenever a space has that property, every

closed

subspace of that space also has the same property.

In the case of a surjection, we can define the following classifications:

A property of a topological space is called

continuous

if it is the case that, whenever a space has this property, the images of this space under all continuous mapping also have the same property.

A property of a topological space is called

open

if it is the case that, whenever a space has this property, the images of this space under all open continuous mappings also have the same property.

A property of a topological space is called

closed invariant

if it is the case that, whenever a space has this property, the images of this space under all closed continuous mapping also have the same property.

Title

classification of topological properties according to behaviour under mapping

Canonical name

ClassificationOfTopologicalPropertiesAccordingToBehaviourUnderMapping

Date of creation

2013-03-22 14:38:04

Last modified on

2013-03-22 14:38:04

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54C05

Defines

hereditary

Defines

hereditarily

Defines

weakly hereditary

Defines

continuous

Defines

open

Defines

closed invariant

Generated on Sat Feb 10 11:17:48 2018 by

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continuity is preserved when codomain is extended

continuity is preserved when codomain is extended

Theorem 1

.

Suppose

X

,

Y

are

topological space

and
let

Z

⊆

Y

be equipped with the

subspace topology

.
If

f

:

X

→

Z

is

continuous

, then

f

:

X

→

Y

is continuous.

Proof.

Let

U

⊆

Y

be an

open set

. Then

f

-

1

⁢

(

U

)

=

{

x

∈

X

:

f

(

x

)

∈

U

}

=

{

x

∈

X

:

f

(

x

)

∈

U

∩

Z

}

=

f

-

1

⁢

(

U

∩

Z

)

.

Since

U

∩

Z

is open in

Z

,

f

-

1

⁢

(

U

)

is open in

X

.
∎

Title

continuity is preserved when codomain is extended

Canonical name

ContinuityIsPreservedWhenCodomainIsExtended

Date of creation

2013-03-22 15:17:14

Last modified on

2013-03-22 15:17:14

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Theorem

Classification

msc 54C05

Related topic

IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous

Generated on Sat Feb 10 11:17:50 2018 by

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continuity of composition of functions

continuity of composition of functions

All functions in this entry are functions from

ℝ

to

ℝ

.

Example 1

Let

f

⁢

(

x

)

=

1

for

x

≤

0

and

f

⁢

(

x

)

=

0

for

x

>

0

,
let

h

⁢

(

x

)

=

0

when

x

∈

ℂ

and

1

when

x

is irrational,
and let

g

⁢

(

x

)

=

h

⁢

(

f

⁢

(

x

)

)

. Then

g

⁢

(

x

)

=

0

for all

x

∈

ℝ

, so
the

composition

of two

discontinuous functions

can be

continuous

.

Example 2

If

g

⁢

(

x

)

=

h

⁢

(

f

⁢

(

x

)

)

is continuous for all functions

f

, then

h

is continuous.
Simply put

f

⁢

(

x

)

=

x

. Same thing for

h

and

f

.
If

g

⁢

(

x

)

=

h

⁢

(

f

⁢

(

x

)

)

is continuous for all functions

h

, then

f

is continuous.
Simply put

h

⁢

(

x

)

=

x

.

Example 3

Suppose

g

⁢

(

x

)

=

h

⁢

(

f

⁢

(

x

)

)

is continuous and

f

is continuous.
Then

h

does not need to be continuous. For a conterexample, put

h

⁢

(

x

)

=

0

for all

x

≠

0

, and

h

⁢

(

0

)

=

1

, and

f

⁢

(

x

)

=

1

+

|

x

|

. Now

h

⁢

(

f

⁢

(

x

)

)

=

0

is continuous, but

h

is not.

Example 4

Suppose

g

⁢

(

x

)

=

h

⁢

(

f

⁢

(

x

)

)

is continuous and

h

is continuous. Then

f

does not
need to be continuous. For a

counterexample

, put

f

⁢

(

x

)

=

0

for all

x

≠

0

, and

f

⁢

(

0

)

=

1

, and

h

⁢

(

x

)

=

0

for all

x

. Now

h

⁢

(

f

⁢

(

x

)

)

=

0

is continuous, but

f

is not.

Title

continuity of composition of functions

Canonical name

ContinuityOfCompositionOfFunctions

Date of creation

2013-03-22 14:04:55

Last modified on

2013-03-22 14:04:55

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Result

Classification

msc 54C05

Classification

msc 26A15

Generated on Sat Feb 10 11:17:52 2018 by

LaTeXML
example of continuous bijections which are not homeomorphisms

example of continuous bijections which are not homeomorphisms

Example 1.

Assume that

X

is a

topological space

, which neither

discrete

nor antidiscrete. We will show that there are topological spaces

Y

and

Z

such that there are

continuous

bijections

X

→

Y

and

Z

→

X

which are not

homeomorphisms

.

Let

Y

=

Z

=

X

as a sets but

topology

on

Y

is antidiscrete and on

Z

is discrete. Then obviously

identity mappings

id

:

X

→

Y

and

id

:

Z

→

X

are continuous, but since

X

is neither discrete nor antidiscrete, these

mappings

are not homeomorphisms.

Example 2.

Consider the

function

f

:

[

0

,

1

)

→

S

1

(here

S

1

denotes the

unit circle

in a

complex plane

) defined by the

formula

f

⁢

(

t

)

=

e

2

⁢

π

⁢

i

⁢

t

. It is easy to see that

f

is a continuous bijection, but

f

is not a homeomorphism (because

[

0

,

1

)

is not

compact

).

Title

example of continuous bijections which are not homeomorphisms

Canonical name

ExampleOfContinuousBijectionsWhichAreNotHomeomorphisms

Date of creation

2013-03-22 18:54:31

Last modified on

2013-03-22 18:54:31

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Example

Classification

msc 54C05

Generated on Sat Feb 10 11:17:54 2018 by

LaTeXML
global versus local continuity

global versus local continuity

In this entry, we establish a very basic fact about

continuity

:

Proposition 1

.

A

function

f

:

X

→

Y

between two

topological spaces

is

continuous

iff it is continuous at every point

x

∈

X

.

Proof.

Suppose first that

f

is continuous, and

x

∈

X

. Let

f

⁢

(

x

)

∈

V

be an

open set

in

Y

. We want to find an open set

x

∈

U

in

X

such that

f

⁢

(

U

)

⊆

V

. Well, let

U

=

f

-

1

⁢

(

V

)

. So

U

is open since

f

is continuous, and

x

∈

U

. Furthermore,

f

⁢

(

U

)

=

f

⁢

(

f

-

1

⁢

(

V

)

)

=

V

.

On the other hand, if

f

is not continuous at

x

∈

X

. Then there is an open set

f

⁢

(

x

)

∈

V

in

Y

such that no open sets

x

∈

U

in

X

have the property

f

⁢

(

U

)

⊆

V

.

(1)

Let

W

=

f

-

1

⁢

(

V

)

. If

W

is open, then

W

has the property

(

1

)

above, a

contradiction

. Since

W

is not open,

f

is not continuous.
∎

Title

global versus local continuity

Canonical name

GlobalVersusLocalContinuity

Date of creation

2013-03-22 19:09:07

Last modified on

2013-03-22 19:09:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Result

Classification

msc 54C05

Classification

msc 26A15

Generated on Sat Feb 10 11:18:04 2018 by

LaTeXML
graph theorems for topological spaces

graph theorems for topological spaces

We wish to show the

relation

between

continuous maps

and their graphs is closer that it may look. Recall, that if

f

:

X

→

Y

is a

function

between sets, then the set

Γ

(

f

)

=

{

(

x

,

f

(

x

)

)

∈

X

×

Y

}

is called

the graph of

f

.

Proposition

1.

If

f

:

X

→

Y

is a continuous map between

topological spaces

such that

Y

is

Hausdorff

, then the graph

Γ

⁢

(

f

)

is a

closed subset

of

X

×

Y

in

product topology

.

Proof.

Indeed, we will show, that

Z

=

(

X

×

Y

)

\

Γ

⁢

(

f

)

is open. Let

(

x

,

y

)

∈

Z

. Then

f

⁢

(

x

)

≠

y

and thus (since

Y

is Hausdorff) there exist open subsetes

V

1

,

V

2

⊆

Y

such that

f

⁢

(

x

)

∈

V

1

,

y

∈

V

2

and

V

1

∩

V

2

=

∅

. Since

f

is

continuous

, then

U

=

f

-

1

⁢

(

V

1

)

is open in

X

.

Note, that the condition

V

1

∩

V

2

=

∅

implies, that

f

⁢

(

U

)

∩

V

2

=

∅

. Therefore

U

×

V

2

is a subset of

Z

. On the other hand this subset is open (since it is a

product

of two

open sets

) in product topology and

(

x

,

y

)

∈

U

×

V

2

. This shows, that every point in

Z

belongs to

Z

together with a small

neighbourhood

, which

completes

the proof.

□

Unfortunetly, the

converse

of this

theorem

is not true as we will see later. Nevertheless we can achieve

similar

result, if we assume a bit more about spaces:

Proposition 2.

Let

f

:

X

→

Y

be a function, where

X

,

Y

are Hausdorff spaces with

Y

compact

. If

Γ

⁢

(

f

)

is a closed subset of

X

×

Y

in product topology, then

f

is continuous.

Proof.

Let

F

⊆

Y

be a

closed set

. We will show that

f

-

1

⁢

(

F

)

is also closed. Consider projections

π

Y

:

X

×

Y

→

Y

;

π

X

:

X

×

Y

→

X

.

They are both continuous and thus

π

Y

-

1

⁢

(

F

)

is closed in

X

×

Y

. Since

Γ

⁢

(

f

)

is also closed, then

Z

=

π

Y

-

1

⁢

(

F

)

∩

Γ

⁢

(

f

)

is closed in

X

×

Y

. It is well known, that since

Y

is compact, then

π

X

is a closed map (this is easily seen to be

equivalent

to

the

tube lemma

). Furthermore it is easy to see, that

π

X

⁢

(

Z

)

=

f

-

1

⁢

(

F

)

and the proof is complete.

□

Counterexample

.

Let

ℝ

denote the set of reals (with

standard topology

). Consider function

f

:

ℝ

→

ℝ

given by

f

⁢

(

x

)

=

1

/

x

and

f

⁢

(

0

)

=

0

. It is obvious, that

f

is

discontinuous

at

x

=

0

, but also it can be easily checked, that

Γ

⁢

(

f

)

is closed in

ℝ

2

. Note, that

ℝ

is not compact.

Title

graph theorems for topological spaces

Canonical name

GraphTheoremsForTopologicalSpaces

Date of creation

2013-03-22 19:15:09

Last modified on

2013-03-22 19:15:09

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Theorem

Classification

msc 54C05

Classification

msc 26A15

Generated on Sat Feb 10 11:18:08 2018 by

LaTeXML
homeomorphism

homeomorphism

A

homeomorphism

f

of

topological spaces

is a

continuous

,

bijective

map such that

f

-

1

is also continuous. We also say that two spaces are

homeomorphic

if such a map exists.

If two topological spaces are homeomorphic, they are

topologically equivalent

— using the techniques of

topology

, there is no way of distinguishing one space from the other.

An

autohomeomorphism

(also known as a

self-homeomorphism

) is a
homeomorphism from a topological space to itself.

Title

homeomorphism

Canonical name

Homeomorphism

Date of creation

2013-03-22 11:59:35

Last modified on

2013-03-22 11:59:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54C05

Synonym

topological equivalence

Synonym

topologically equivalent

Related topic

Homeotopy

Related topic

CrosscapSlide

Related topic

AlexanderTrick

Related topic

GroupoidCategory

Defines

homeomorphic

Defines

autohomeomorphism

Defines

auto-homeomorphism

Defines

self-homeomorphism

Generated on Sat Feb 10 11:18:10 2018 by

LaTeXML
injective map between real numbers is a homeomorphism

injective map between real numbers is a homeomorphism

Lemma.

Assume that

I

is an

open interval

and

f

:

I

→

ℝ

is an

injective

,

continuous map

. Then

f

⁢

(

I

)

⊆

ℝ

is an

open subset

.

Proof.

Since

f

is injective, then of course

f

is

monotonic

. Without loss of generality, we may assume that

f

is increasing. Let

y

=

f

⁢

(

x

)

∈

f

⁢

(

I

)

. Since

I

is open, then there are

α

,

β

∈

I

such that

α

<

x

<

β

. Therefore

f

⁢

(

α

)

<

y

<

f

⁢

(

β

)

and (because

continuous functions

are Darboux functions) for any

y

′

∈

(

f

⁢

(

α

)

,

f

⁢

(

β

)

)

there exists

x

′

∈

I

such that

f

⁢

(

x

′

)

=

y

′

. This shows that

(

f

⁢

(

α

)

,

f

⁢

(

β

)

)

is an open

neighbourhood

of

y

contained

in

f

⁢

(

I

)

and therefore (since

y

was arbitrary)

f

⁢

(

I

)

is open.

□

Proposition

.

Assume that

I

is an open interval and

f

:

I

→

ℝ

is an injective, continuous map. Then

f

is a

homeomorphism

onto image.

Proof.

Of course, it is enough to show that

f

is an open map. But if

U

⊆

I

is open, then there are

disjoint

, open intervals

I

α

such that

U

=

⋃

α

I

α

.

Therefore we obtain

continuous

, injective maps

f

α

:

I

α

→

ℝ

which are

restrictions

of

f

to

I

α

. By lemma we have that

f

α

⁢

(

I

α

)

is open and therefore

f

⁢

(

U

)

=

f

⁢

(

⋃

α

I

α

)

=

⋃

α

f

⁢

(

I

α

)

=

⋃

α

f

α

⁢

(

I

α

)

is open. This shows that

f

is a homeomorphism onto image.

□

Title

injective map between

real numbers

is a homeomorphism

Canonical name

InjectiveMapBetweenRealNumbersIsAHomeomorphism

Date of creation

2013-03-22 18:53:58

Last modified on

2013-03-22 18:53:58

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 54C05

Generated on Sat Feb 10 11:18:12 2018 by

LaTeXML
inverse function theorem (topological spaces)

inverse function theorem (topological spaces)

Let

X

and

Y

be

topological spaces

, with

X

compact

and

Y

Hausdorff

. Suppose

f

:

X

→

Y

is a

continuous

bijection

. Then

f

is a

homeomorphism

, i.e.

f

-

1

is continuous.

Note if

Y

is a

metric space

, then it is Hausdorff, and the

theorem

holds.

Title

inverse function theorem

(topological spaces)

Canonical name

InverseFunctionTheoremtopologicalSpaces

Date of creation

2013-03-22 13:25:04

Last modified on

2013-03-22 13:25:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 54C05

Related topic

Compact

Generated on Sat Feb 10 11:18:13 2018 by

LaTeXML
local homeomorphism

local homeomorphism

Definition.

Let

X

and

Y

be

topological spaces

.

Continuous map

f

:

X

→

Y

is said to be

locally

invertible

in

x

∈

X

iff there exist

open subsets

U

⊆

X

and

V

⊆

Y

such that

x

∈

U

,

f

⁢

(

x

)

∈

V

and the

restriction

f

:

U

→

V

is a

homeomorphism

. If

f

is locally invertible in every point of

X

, then

f

is called a

local homeomorphism

.

Examples.

Of course every homeomorphism is a local homeomorphism, but the

converse

is not true. For example, let

f

:

ℂ

→

ℂ

be an

exponential function

, i.e.

f

⁢

(

z

)

=

e

z

. Then

f

is a local homeomorphism, but it is not a homeorphism (indeed,

f

⁢

(

z

)

=

f

⁢

(

z

+

2

⁢

π

⁢

i

)

for any

z

∈

ℂ

).

One of the most important

theorem

of

differential calculus

(i.e.

inverse function theorem

) states, that if

f

:

M

→

N

is a

C

1

-map between

C

1

-manifolds such that

T

x

⁢

f

:

T

x

⁢

M

→

T

f

⁢

(

x

)

⁢

N

is a linear isomorphism for a given

x

∈

M

, then

f

is locally invertible in

x

(in this case the local

inverse

is even a

C

1

-map).

Title

local homeomorphism

Canonical name

LocalHomeomorphism

Date of creation

2013-03-22 18:53:47

Last modified on

2013-03-22 18:53:47

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 54C05

Generated on Sat Feb 10 11:18:15 2018 by

LaTeXML
local homeomorphisms between real numbers

local homeomorphisms between real numbers

Proposition

.

Let

I

be an

open interval

and

f

:

I

→

ℝ

be a

continuous map

. Then

f

is a

local homeomorphism

if and only if

f

is a

homeomorphism

onto image.

Proof.

,,

⇐

” If

f

is a homeomorphism onto image, then (in particular)

f

is

monotonic

and

continuous

, thus

f

⁢

(

I

)

is open in

ℝ

(please, see this entry (

http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism

) for more details). It is easy to see that therefore

f

is a local homeomorphism.

,,

⇒

” Assume that

f

is not a homeomorphism onto image. It is well known, that this implies that

f

is not

injective

(please, see this entry (

http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism

) for more details). Let

x

,

y

∈

I

be such that

x

<

y

and

f

⁢

(

x

)

=

f

⁢

(

y

)

. Then there exists

c

∈

I

such that

x

<

c

<

y

and

c

is a

local maximum

of

f

. Thus (since

f

is a Darboux function) for any

ε

>

0

there are points

x

ε

,

y

ε

∈

(

c

-

ε

,

c

+

ε

)

such that

f

⁢

(

x

ε

)

=

f

⁢

(

y

ε

)

. This obviously implies that

f

cannot be locally inverted around

c

.

□

Title

local homeomorphisms between real numbers

Canonical name

LocalHomeomorphismsBetweenRealNumbers

Date of creation

2013-03-22 18:53:50

Last modified on

2013-03-22 18:53:50

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Theorem

Classification

msc 54C05

Generated on Sat Feb 10 11:18:17 2018 by

LaTeXML