aarjaneiro/planetmath huggingface dataset:
connected space

connected space

A

topological space

X

is said to be

connected

if there is no pair of nonempty subsets

U

,

V

such that both

U

and

V

are open in

X

,

U

∩

V

=

∅

and

U

∪

V

=

X

. If

X

is not connected, i.e. if there are sets

U

and

V

with the above properties, then we say that

X

is

disconnected

.

Every topological space

X

can be viewed as a

collection

of

subspaces

each of which are connected. These subspaces are called the

connected components

of

X

. Slightly more rigorously, we define an

equivalence relation

∼

on points in

X

by declaring that

x

∼

y

if there is a connected subset

Y

of

X

such that

x

and

y

both lie in

Y

. Then a connected component of

X

is defined to be an

equivalence class

under this

relation

.

Title

connected space

Canonical name

ConnectedSpace

Date of creation

2013-03-22 12:00:11

Last modified on

2013-03-22 12:00:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54D05

Related topic

SemilocallySimplyConnected

Related topic

ExtremallyDisconnected

Related topic

ExampleOfAConnectedSpaceWhichIsNotPathConnected

Related topic

LocallyConnected

Related topic

ProofOfGeneralizedIntermediateValueTheorem

Related topic

AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2

Related topic

AConnectedNormalSpaceWithMoreThanOn

Defines

connected

Defines

connected components

Defines

disconnected

Defines

connectedness

Generated on Sat Feb 10 11:20:28 2018 by

LaTeXML
connectedness is preserved under a continuous map

connectedness is preserved under a continuous map

Theorem

Suppose

f

:

X

→

Y

is a

continuous map

between

topological spaces

X

and

Y

.
If

X

is a

connected space

, and

f

is

surjective

,
then

Y

is a connected space.

The

inclusion map

for spaces

X

=

(

0

,

1

)

and

Y

=

(

0

,

1

)

∪

(

2

,

3

)

shows
that we need to assume that the map is surjective. Othewise,
we can only prove that

f

⁢

(

X

)

is

connected

.
See this page (

http://planetmath.org/IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous

).

Proof.

For a

contradiction

, suppose there
are

disjoint

open sets

A

,

B

in

Y

such that

Y

=

A

∪

B

.
By

continuity

and properties of the

inverse image

,

f

-

1

⁢

(

A

)

and

f

-

1

⁢

(

B

)

are open

disjoint sets

in

X

.
Since

f

is surjective,

Y

=

f

⁢

(

X

)

=

A

∪

B

, whence

X

=

f

-

1

⁢

f

⁢

(

X

)

=

f

-

1

⁢

(

A

)

∪

f

-

1

⁢

(

B

)

contradicting the

assumption

that

X

is connected.

References

1

G.J. Jameson,

Topology

and

Normed Spaces

,
Chapman and Hall, 1974.

2

G.L. Naber,

Topological methods in

Euclidean spaces

,
Cambridge University Press, 1980.

Title

connectedness is preserved under a continuous map

Canonical name

ConnectednessIsPreservedUnderAContinuousMap

Date of creation

2013-03-22 13:55:59

Last modified on

2013-03-22 13:55:59

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Theorem

Classification

msc 54D05

Related topic

CompactnessIsPreservedUnderAContinuousMap

Related topic

ProofOfGeneralizedIntermediateValueTheorem

Generated on Sat Feb 10 11:20:30 2018 by

LaTeXML
continuous images of path connected spaces are path connected

continuous images of path connected spaces are path connected

Proposition

.

The

continuous

image of a

path connected

space is path connected.

Proof.

Let

X

be a path connected space, and suppose

f

is a
continuous

surjection

whose domain is

X

. Let

a

and

b

be points in the image of

f

. Each has at least one

preimage

in

X

, and by the path

connectedness

of

X

, there is a path in

X

from a preimage of

a

to a preimage of

b

. Applying

f

to this path yields a path in the image of

f

from

a

to

b

.
∎

Title

continuous images of path connected spaces are path connected

Canonical name

ContinuousImagesOfPathConnectedSpacesArePathConnected

Date of creation

2013-03-22 15:52:38

Last modified on

2013-03-22 15:52:38

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Result

Classification

msc 54D05

Generated on Sat Feb 10 11:20:31 2018 by

LaTeXML
cut-point

cut-point

Theorem

Suppose

X

is a

connected space

and

x

is a point in

X

.
If

X

∖

{

x

}

is a disconnected set in

X

, then

x

is a

cut-point

of

X

[

1

,

2

]

.

0.0.1

Examples

1.

Any point of

ℝ

with the

usual topology

is a cut-point.

2.

If

X

is a

normed vector space

with

dim

⁡

X

>

1

, then

X

has
no cut-points

[

1

]

.

References

1

G.J. Jameson,

Topology

and

Normed Spaces

,
Chapman and Hall, 1974.

2

L.E. Ward,

Topology, An Outline for a First Course

,
Marcel Dekker, Inc., 1972.

Title

cut-point

Canonical name

Cutpoint

Date of creation

2013-03-22 13:56:38

Last modified on

2013-03-22 13:56:38

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54D05

Synonym

cutpoint

Generated on Sat Feb 10 11:20:33 2018 by

LaTeXML
example of a connected space that is not path-connected

example of a connected space that is not path-connected

This standard example shows that
a

connected

topological space

need not be

path-connected

(the

converse

is true, however).

Consider the topological spaces

X

1

=

{

(

0

,

y

)

∣

y

∈

[

-

1

,

1

]

}

X

2

=

{

(

x

,

sin

⁡

1

x

)

⁢

∣

x

>

⁢

0

}

X

=

X

1

∪

X

2

with the

topology

induced

from

ℝ

2

.

X

2

is often called the “

topologist’s sine curve

”, and

X

is its

closure

.

X

is not path-connected.
Indeed, assume to the contrary that there exists a path (

http://planetmath.org/PathConnected

)

γ

:

[

0

,

1

]

→

X

with

γ

⁢

(

0

)

=

(

1

π

,

0

)

and

γ

⁢

(

1

)

=

(

0

,

0

)

.
Let

c

=

inf

⁡

{

t

∈

[

0

,

1

]

∣

γ

⁢

(

t

)

∈

X

1

}

.

Then

γ

⁢

(

[

0

,

c

]

)

contains at most one point of

X

1

,
while

γ

⁢

(

[

0

,

c

]

)

¯

contains all of

X

1

.
So

γ

⁢

(

[

0

,

c

]

)

is not closed, and therefore not

compact

.
But

γ

is

continuous

and

[

0

,

c

]

is compact,
so

γ

⁢

(

[

0

,

c

]

)

must be compact
(as a continuous image of a compact set is compact),
which is a

contradiction

.

But

X

is connected.
Since both “parts” of the topologist’s sine curve are themselves connected,
neither can be partitioned into two

open sets

.
And any open set which contains points of the

line segment

X

1

must contain points of

X

2

.
So

X

is not the

disjoint union

of two nonempty open sets,
and is therefore connected.

Title

example of a connected space that is not path-connected

Canonical name

ExampleOfAConnectedSpaceThatIsNotPathconnected

Date of creation

2013-03-22 12:46:33

Last modified on

2013-03-22 12:46:33

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

16

Author

yark (2760)

Entry type

Example

Classification

msc 54D05

Related topic

ConnectedSpace

Related topic

PathConnected

Defines

topologist’s sine curve

Generated on Sat Feb 10 11:20:35 2018 by

LaTeXML
example of a semilocally simply connected space which is not locally simply connected

example of a semilocally simply connected space which is not locally simply connected

Let

H

⁢

R

be the

Hawaiian rings

, and define

X

to be the cone over

H

⁢

R

.

Then,

X

is

connected

,

locally connected

, and

semilocally simply connected

, but

not

locally simply connected

.

Too see this, let

p

∈

H

⁢

R

be the point to which the circles

converge

in

H

⁢

R

,

and represent

X

as

H

⁢

R

×

[

0

,

1

]

/

H

⁢

R

×

{

0

}

.

Then, every small enough

neighborhood

of

q

:=

(

p

,

1

)

∈

X

fails to be

simply connected

. However, since

X

is a cone, it is

contractible

, so all loops (in particular, loops in a neighborhood of

q

) can be contracted to a point within

X

.

Title

example of a semilocally simply connected space which is not locally simply connected

Canonical name

ExampleOfASemilocallySimplyConnectedSpaceWhichIsNotLocallySimplyConnected

Date of creation

2013-03-22 13:25:15

Last modified on

2013-03-22 13:25:15

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

5

Author

antonio (1116)

Entry type

Example

Classification

msc 54D05

Classification

msc 57M10

Generated on Sat Feb 10 11:20:37 2018 by

LaTeXML
homeomorphisms preserve connected components

homeomorphisms preserve connected components

Let

X

,

Y

be

topological spaces

and

X

=

⋃

X

i

,

Y

=

⋃

Y

j

be decompositions into

connected components

.

Proposition

.

Assume that

f

:

X

→

Y

is a

homeomorphism

. Then for any

i

there exists

j

such that

f

⁢

(

X

i

)

=

Y

j

.

Proof.

Take any

i

. Because

f

is

continuous

f

⁢

(

X

i

)

is

connected

, then there exists

j

such that

f

⁢

(

X

i

)

⊆

Y

j

(because

Y

j

is a connected component). Now

f

is a homeomorphism,

f

-

1

⁢

(

Y

j

)

∩

X

i

≠

∅

,

Y

j

is connected and

X

i

is a connected component, so

f

-

1

⁢

(

Y

j

)

⊆

X

i

. Thus

Y

j

⊆

f

⁢

(

X

i

)

, which

completes

the proof.

□

Title

homeomorphisms preserve connected components

Canonical name

HomeomorphismsPreserveConnectedComponents

Date of creation

2013-03-22 18:45:32

Last modified on

2013-03-22 18:45:32

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Derivation

Classification

msc 54D05

Generated on Sat Feb 10 11:20:38 2018 by

LaTeXML
hyperconnected space

hyperconnected space

A

topological space

X

is said to be

hyperconnected

if no pair of nonempty

open sets

of

X

is

disjoint

(or, equivalently, if

X

is not the union of two proper

closed sets

).
Hyperconnected spaces are sometimes known as

irreducible sets

(

http://planetmath.org/IrreducibleClosedSet

).

All hyperconnected spaces are

connected

,

locally connected

, and

pseudocompact

.

Any

infinite set

with the

cofinite topology

is an example of a hyperconnected space.
Similarly, any

uncountable set

with the cocountable topology is hyperconnected.

Affine spaces

and

projectives spaces

over an

infinite

field, when endowed with the

Zariski topology

, are also hyperconnected.

Title

hyperconnected space

Canonical name

HyperconnectedSpace

Date of creation

2013-03-22 14:20:30

Last modified on

2013-03-22 14:20:30

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 54D05

Synonym

hyper-connected space

Related topic

UltraconnectedSpace

Related topic

IrreducibleClosedSet

Defines

hyperconnected

Defines

hyper-connected

Generated on Sat Feb 10 11:20:40 2018 by

LaTeXML
Jordan curve theorem

Jordan curve theorem

Informally, the

Jordan curve theorem

states that every

Jordan curve

divides the

Euclidean plane

into an “outside” and an “inside”. The proof of this geometrically plausible result requires surprisingly heavy machinery from

topology

. The difficulty lies in the great generality of the statement and inherent difficulty in formalizing the exact meaning of words like “curve”, “inside”, and “outside.”

There are several

equivalent

formulations.

Theorem 1

.

If

Γ

is a simple closed curve in

R

2

, then

R

2

∖

Γ

has precisely two

connected components

(

http://planetmath.org/ConnectedSpace

).

Theorem 2

.

If

Γ

is a simple closed curve in the sphere

S

2

, then

S

2

∖

Γ

consists of precisely two connected components.

Theorem 3

.

Let

h

:

R

→

R

2

be a one-to-one

continuous map

such that

|

h

⁢

(

t

)

|

→

∞

as

|

t

|

→

∞

. Then

R

2

∖

h

⁢

(

R

)

consists of precisely two connected components.

The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in. There we can define “inside” to be the

bounded

connected component, as any picture can easily convey.

Title

Jordan curve theorem

Canonical name

JordanCurveTheorem

Date of creation

2013-03-22 13:08:53

Last modified on

2013-03-22 13:08:53

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Theorem

Classification

msc 54D05

Classification

msc 54A05

Generated on Sat Feb 10 11:20:44 2018 by

LaTeXML
limit points and closure for connected sets

limit points and closure for connected sets

The below

theorem

shows that adding

limit points

to a connected
set preserves

connectedness

.

Theorem 1

.

Suppose

A

is a connected set in a

topological space

.
If

A

⊆

B

⊆

A

¯

, then

B

is connected.
In particular,

A

¯

is connected.

Thus, one way to prove that a space

X

is connected is to find a dense
subspace in

X

which is connected.

Two touching

closed balls

in

ℝ

2

shows that this theorem does not hold
for the

interior

. Along the same lines, taking the

closure

does not
preserve separatedness.

Proof.

Let

X

be the ambient topological space.
By

assumption

, if

U

,

V

⊆

A

are open and

U

∪

V

=

A

, then

U

∩

V

≠

∅

.
To prove that

B

is connected, let

U

,

V

be

open sets

in

B

such that

U

∪

V

=

B

and for a
contradition, suppose that

U

∩

V

=

∅

.
Then there are open sets

R

,

S

⊆

X

such that

U

=

R

∩

B

,

V

=

S

∩

B

.

It follows that

(

R

∪

S

)

∩

B

=

B

and

(

R

∩

S

)

∩

B

=

∅

.
Next, let

U

~

,

V

~

be open sets in

A

defined as

U

~

=

R

∩

A

,

V

~

=

S

∩

A

.

Now

A

=

B

∩

A

⊆

(

R

∪

S

)

∩

A

⊆

A

and as

(

R

∪

S

)

∩

A

=

U

~

∪

V

~

, it follows that

∅

≠

U

~

∩

V

~

=

(

R

∩

S

)

∩

A

.
Then, by the properties of the

closure operator

,

∅

≠

(

R

∩

S

)

∩

A

¯

⊇

(

R

∩

S

)

∩

A

¯

⊇

(

R

∩

S

)

∩

B

=

∅

.

∎

Title

limit points and closure for connected sets

Canonical name

LimitPointsAndClosureForConnectedSets

Date of creation

2013-03-22 15:17:56

Last modified on

2013-03-22 15:17:56

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 54D05

Generated on Sat Feb 10 11:20:47 2018 by

LaTeXML
locally connected

locally connected

A

topological space

X

is

locally connected

at a point

x

∈

X

if every

neighborhood

U

of

x

contains a

connected

neighborhood

V

of

x

. The space

X

is

locally connected

if it is locally connected at every point

x

∈

X

.

A topological space

X

is

locally path connected

at a point

x

∈

X

if every neighborhood

U

of

x

contains a

path connected

neighborhood

V

of

x

. The space

X

is

locally path connected

if it is locally path connected at every point

x

∈

X

.

Title

locally connected

Canonical name

LocallyConnected

Date of creation

2013-03-22 12:38:48

Last modified on

2013-03-22 12:38:48

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 54D05

Related topic

ConnectedSet

Related topic

ConnectedSpace

Related topic

PathConnected

Related topic

SemilocallySimplyConnected

Defines

locally path connected

Generated on Sat Feb 10 11:20:48 2018 by

LaTeXML
locally simply connected

locally simply connected

Let

X

be a

topological space

and

x

∈

X

.

X

is said to be

locally simply connected

at

x

, if every

neighborhood

of

x

contains a

simply connected

neighborhood of

x

.

X

is said to be locally simply connected if it is locally simply connected at every point.

Title

locally simply connected

Canonical name

LocallySimplyConnected

Date of creation

2013-03-22 13:25:23

Last modified on

2013-03-22 13:25:23

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

6

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 54D05

Synonym

locally 1-connected

Defines

locally simply connected

Generated on Sat Feb 10 11:20:50 2018 by

LaTeXML
path

path

Let

I

=

[

0

,

1

]

⊂

ℝ

and let

X

be a

topological space

.

A

continuous map

f

:

I

→

X

such that

f

⁢

(

0

)

=

x

and

f

⁢

(

1

)

=

y

is called a

path

in

X

. The point

x

is called the

initial point

of the path and

y

is called its

terminal point

. If, in

addition

, the map is

one-to-one

, then it is known as an

arc

.

Sometimes, it is convenient to regard two paths or arcs as

equivalent

if they differ by a reparameterization. That is to say, we regard

f

:

I

→

X

and

g

:

I

→

X

as equivalent if there exists a

homeomorphism

h

:

I

→

I

such that

h

⁢

(

0

)

=

0

and

h

⁢

(

1

)

=

1

and

f

=

g

∘

h

.

If the space

X

has extra

structure

, one may choose to restrict the classes of paths and reparameterizations. For example, if

X

has a differentiable structure, one may consider the class of

differentiable

paths. Likewise, one can speak of

piecewise

linear paths,

rectifiable paths

, and

analytic

paths in suitable

contexts

.

The space

X

is said to be

pathwise connected

if, for every two points

x

,

y

∈

X

, there exists a path having

x

as initial point and

y

as terminal point. Likewise, the space

X

is said to be

arcwise connected

if, for every two distinct points

x

,

y

∈

X

, there exists an

arc

having

x

as initial point and

y

as terminal point.

A pathwise connected space is always a

connected space

, but a connected space need not be path connected. An arcwise connected space is always a pathwise connected space, but a pathwise connected space need not be arcwise connected.
As it turns out, for

Hausdorff spaces

these two notions coincide — a Hausdorff space is pathwise connected iff it is arcwise connected.

Title

path

Canonical name

Path

Date of creation

2013-03-22 12:00:15

Last modified on

2013-03-22 12:00:15

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54D05

Synonym

pathwise connected

Synonym

path-connected

Synonym

path connected

Related topic

SimplePath

Related topic

DistanceInAGraph

Related topic

LocallyConnected

Related topic

ExampleOfAConnectedSpaceWhichIsNotPathConnected

Related topic

PathConnectnessAsAHomotopyInvariant

Defines

path

Defines

arc

Defines

arcwise connected

Defines

initial point

Defines

terminal point

Generated on Sat Feb 10 11:20:51 2018 by

LaTeXML
path component

path component

Two points

x

,

y

in a

topological space

X

are said to be in the same

path component

if there exists a path from

x

to

y

in

X

. The

equivalence classes

of

X

under this

equivalence relation

are called the

path components

of

X

.

Title

path component

Canonical name

PathComponent

Date of creation

2013-03-22 12:00:24

Last modified on

2013-03-22 12:00:24

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 54D05

Generated on Sat Feb 10 11:20:53 2018 by

LaTeXML
product of path connected spaces is path connected

product of path connected spaces is path connected

Proposition

. Let

X

and

Y

be

topological spaces

. Then

X

×

Y

is

path connected

if and only if both

X

and

Y

are path connected.

Proof

. ”

⇐

” Assume that

X

and

Y

are path connected and let

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

∈

X

×

Y

be arbitrary points. Since

X

is path connected, then there exists a continous map

σ

:

I

→

X

such that

σ

⁢

(

0

)

=

x

1

and

σ

⁢

(

1

)

=

x

2

.

Analogously there exists a continous map

τ

:

I

→

Y

such that

τ

⁢

(

0

)

=

y

1

and

τ

⁢

(

1

)

=

y

2

.

Then we have an

induced map

σ

×

τ

:

I

→

X

×

Y

defined by the

formula

:

(

σ

×

τ

)

⁢

(

t

)

=

(

σ

⁢

(

t

)

,

τ

⁢

(

t

)

)

,

which is continous path from

(

x

1

,

y

1

)

to

(

x

2

,

y

2

)

.

”

⇒

” On the other hand assume that

X

×

Y

is path connected. Let

x

1

,

x

2

∈

X

and

y

0

∈

Y

. Then there exists a path

σ

:

I

→

X

×

Y

such that

σ

⁢

(

0

)

=

(

x

1

,

y

0

)

and

σ

⁢

(

1

)

=

(

x

1

,

y

0

)

.

We also have the

projection map

π

:

X

×

Y

→

X

such that

π

⁢

(

x

,

y

)

=

x

. Thus we have a map

τ

:

I

→

X

defined by the formula

τ

⁢

(

t

)

=

π

⁢

(

σ

⁢

(

t

)

)

.

This is a continous path from

x

1

to

x

2

, therefore

X

is path connected. Analogously

Y

is path connected.

□

Title

product of path connected spaces is path connected

Canonical

name

ProductOfPathConnectedSpacesIsPathConnected

Date of creation

2013-03-22 18:31:02

Last modified on

2013-03-22 18:31:02

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 54D05

Generated on Sat Feb 10 11:20:55 2018 by

LaTeXML
products of connected spaces are connected

products of connected spaces are connected

Theorem 1

[

1

,

2

]

Let

(

X

i

)

i

∈

I

be a family of

topological spaces

.
Then the product space

∏

i

∈

I

X

i

with the

product topology

is

connected

if and only if each
space

X

i

is connected.

As is true of most results in

topology

involving

products

,
the forward

implication

requires the

axiom of choice

.

References

1

S. Lang,

Analysis

II

,
Addison-Wesley Publishing Company Inc., 1969.

2

A. Mukherjea, K. Pothoven,

Real and

Functional Analysis

,
Plenum Press, 1978.

Title

products of connected spaces are connected

Canonical name

ProductsOfConnectedSpacesAreConnected

Date of creation

2013-03-22 13:56:13

Last modified on

2013-03-22 13:56:13

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Theorem

Classification

msc 54D05

Generated on Sat Feb 10 11:20:57 2018 by

LaTeXML
proof that a path connected space is connected

proof that a path connected space is connected

Let

X

be a

path connected

topological space

. Suppose that

X

=

A

∪

B

, where

A

and

B

are non empty,

disjoint

,

open sets

. Let

a

∈

A

,

b

∈

B

, and let

γ

:

I

→

X

denote a path from

a

to

b

.

We have

I

=

γ

-

1

⁢

(

A

)

∪

γ

-

1

⁢

(

B

)

, where

γ

-

1

⁢

(

A

)

,

γ

-

1

⁢

(

B

)

are non empty, open and disjoint. Since

I

is

connected

, this is a

contradiction

, which concludes the proof.

Title

proof that a path connected space is connected

Canonical name

ProofThatAPathConnectedSpaceIsConnected

Date of creation

2013-03-22 12:46:30

Last modified on

2013-03-22 12:46:30

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

6

Author

n3o (216)

Entry type

Proof

Classification

msc 54D05

Generated on Sat Feb 10 11:20:58 2018 by

LaTeXML