aarjaneiro/planetmath huggingface dataset:
proof of inverse function theorem (topological spaces)

proof of inverse function theorem (topological spaces)

We only have to prove that whenever

A

⊂

X

is an

open set

, then also

B

=

(

f

-

1

)

-

1

⁢

(

A

)

=

f

⁢

(

A

)

⊂

Y

is open (

f

is an

open mapping

). Equivalently it is enough to prove that

B

′

=

Y

∖

B

is closed.

Since

f

is

bijective

we have

B

′

=

Y

∖

B

=

f

⁢

(

X

∖

A

)

As

A

′

=

X

∖

A

is closed and since

X

is

compact

A

′

is compact too (this and the following are well know

properties of compact spaces

).
Moreover being

f

continuous

we know that also

B

′

=

f

⁢

(

A

′

)

is compact. Finally since

Y

is

Hausdorff

then

B

′

is closed.

Title

proof of inverse function theorem

(

topological spaces

)

Canonical name

ProofOfInverseFunctionTheoremtopologicalSpaces

Date of creation

2013-03-22 13:31:55

Last modified on

2013-03-22 13:31:55

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Proof

Classification

msc 54C05

Generated on Sat Feb 10 11:18:19 2018 by

LaTeXML
restriction of a continuous mapping is continuous

restriction of a continuous mapping is continuous

Theorem

Suppose

X

and

Y

are

topological spaces

, and suppose

f

:

X

→

Y

is a

continuous function

. For a subset

A

⊂

X

,
the

restriction

(

http://planetmath.org/RestrictionOfAFunction

)
of

f

to

A

(that is

f

|

A

) is a continuous
mapping

f

|

A

:

A

→

Y

, where

A

is given the

subspace topology

from

X

.

Proof.

We need to show that for any

open set

V

⊂

Y

, we
can write

(

f

|

A

)

-

1

⁢

(

V

)

=

A

∩

U

for some set

U

that is open in

X

.
However, by the properties of the

inverse image

(see
this page (

http://planetmath.org/InverseImage

)), we have for any open set

V

⊂

Y

,

(

f

|

A

)

-

1

⁢

(

V

)

=

A

∩

f

-

1

⁢

(

V

)

.

Since

f

:

X

→

Y

is

continuous

,

f

-

1

⁢

(

V

)

is open in

X

, and
our claim follows.

□

Title

restriction of a continuous mapping is continuous

Canonical name

RestrictionOfAContinuousMappingIsContinuous

Date of creation

2013-03-22 13:55:53

Last modified on

2013-03-22 13:55:53

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Theorem

Classification

msc 54C05

Classification

msc 26A15

Generated on Sat Feb 10 11:18:20 2018 by

LaTeXML
sequentially continuous

sequentially continuous

Let

X

,

Y

be

topological spaces

. Then a

f

:

X

→

Y

is said to be

sequentially continuous

if for every

convergent sequence

x

n

→

x

in

X

,

f

⁢

(

x

n

)

→

f

⁢

(

x

)

in

Y

.

Every

continuous function

is sequentially continuous, however the

converse

is true only in

first-countable spaces

(for example in

metric spaces

).

Title

sequentially continuous

Canonical name

SequentiallyContinuous

Date of creation

2013-03-22 16:31:17

Last modified on

2013-03-22 16:31:17

Owner

ehremo (15714)

Last modified by

ehremo (15714)

Numerical id

6

Author

ehremo (15714)

Entry type

Definition

Classification

msc 54C05

Generated on Sat Feb 10 11:18:22 2018 by

LaTeXML
proximity continuous

proximity continuous

Let

(

P

,

δ

)

and

(

Q

,

ϵ

)

be

proximity spaces

.

A

function

f

:

P

→

Q

is said to be

proximity continuous

if for any subsets

A

,

B

⊆

P

,

A

𝛿

B

implies that

f

⁢

(

A

)

ϵ

f

⁢

(

B

)

.

Title

proximity continuous

Canonical name

ProximityContinuous

Date of creation

2013-03-22 16:56:12

Last modified on

2013-03-22 16:56:12

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

12

Author

porton (9363)

Entry type

Definition

Classification

msc 54C08

Classification

msc 54C05

Classification

msc 54E17

Classification

msc 54E05

Synonym

near-continuousness

Defines

δ

-

continuous

Generated on Sat Feb 10 11:18:33 2018 by

LaTeXML
uniformly continuous is proximity continuous

uniformly continuous is proximity continuous

Any

uniformly continuous

mapping is

proximity continuous

regarding the

proximities

generated by the

uniformities

(

http://planetmath.org/ProximityGeneratedByUniformity

).

Title

uniformly continuous is proximity continuous

Canonical name

UniformlyContinuousIsProximityContinuous

Date of creation

2013-03-22 16:56:15

Last modified on

2013-03-22 16:56:15

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

6

Author

porton (9363)

Entry type

Theorem

Classification

msc 54C08

Classification

msc 54C05

Classification

msc 54E17

Classification

msc 54E05

Classification

msc 54E15

Generated on Sat Feb 10 11:18:35 2018 by

LaTeXML
crosscap slide

crosscap slide

The

crosscap slide

is an

autohomeomorphism

which can be defined only for

non orientable surfaces

whose genus is greater than one.

It is also known as the y-homeomorphism. This name was coined by Lickorish and the name crosscap slide was introduced by Korkmaz.

1.

D.R.J. Chillingworth.

A

finite set

of

generators

for the
homeotopy group of a non-orientable surface

, Proc. Camb. Phil. Soc.
65(1969), 409-430.

2.

M. Korkmaz.

Mapping Class Groups

of Non-orientable Surfaces

, Geometriae Dedicata 89 (2002), 109-133.

3.

W.B.R. Lickorish.

Homeomorphisms of non-orientable two-manifolds

,
Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.

Title

crosscap slide

Canonical name

CrosscapSlide

Date of creation

2013-03-22 15:42:23

Last modified on

2013-03-22 15:42:23

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

5

Author

juanman (12619)

Entry type

Definition

Classification

msc 54C10

Synonym

y-homeomorphism

Related topic

homeomorphism

Related topic

YHomeomorphism2

Related topic

Homeomorphism

Generated on Sat Feb 10 11:18:36 2018 by

LaTeXML
open mapping

open mapping

Let

X

and

Y

be two

topological spaces

. A function

f

:

X

→

Y

is
said to be open if

f

⁢

(

U

)

is open for each

open subset

U

of

X

.

Accordingly, if

f

⁢

(

C

)

is closed for each

closed subset

C

of

X

, we say that

f

is closed.

Title

open mapping

Canonical name

OpenMapping

Date of creation

2013-03-22 13:12:45

Last modified on

2013-03-22 13:12:45

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 54C10

Synonym

open function

Defines

closed function

Defines

closed mapping

Generated on Sat Feb 10 11:18:38 2018 by

LaTeXML
proper map

proper map

Definition

Suppose

X

and

Y

are

topological spaces

, and

f

is a map

f

:

X

→

Y

.
Then

f

is a

proper map

if the

inverse image

of every

compact
subset

in

Y

of is a compact set in

X

.

Title

proper map

Canonical name

ProperMap

Date of creation

2013-03-22 13:59:49

Last modified on

2013-03-22 13:59:49

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Definition

Classification

msc 54C10

Classification

msc 54-00

Related topic

PolynomialFunctionIsAProperMap

Generated on Sat Feb 10 11:18:40 2018 by

LaTeXML
surjective open maps in terms of nets

surjective open maps in terms of nets

Theorem -

Let

f

:

X

⟶

Y

be a surjective map between the

topological spaces

X

and

Y

. Then

f

is an

open mapping

if and only if given a net

{

y

i

}

i

∈

I

⊂

Y

such that

y

i

⟶

y

, then for every

x

∈

f

-

1

⁢

(

{

y

}

)

there exists a subnet

{

y

i

j

}

j

∈

J

that

Generated on Sat Feb 10 11:18:41 2018 by

LaTeXML
y-homeomorphism

y-homeomorphism

The

y-homeomorphism

also dubbed

crosscap slide

, is an

auto-homeomorphism

(or self-homeomorphism) which can be defined only for

non orientable surfaces

whose genus is greater than one.

To define it we take a punctured

Klein bottle

K

0

=

K

∖

int

⁢

D

2

which can be consider as a pair of closed Möbius bands

M

1

,

M

2

, one sewed in the other by perforating with a disk (being disjoint from

∂

⁡

M

1

) and then identify the

boundary

of the second with the boundary of that disk, in symbols:

K

0

=

(

M

1

∖

int

⁢

D

2

)

∪

∂

M

2

where

∂

=

∂

⁡

D

2

=

∂

⁡

M

2

. Other way to visualizing that, is by consider

K

0

as the

connected sum

of

int

⁢

M

1

with a

projective plane

ℝ

⁢

P

2

.

Now, thinking that the removed disk

D

2

was located with its center at some point in the core of

M

1

, the second band,

M

2

will have a pair of points on that part of the core in common with

∂

⁡

M

2

.

So, the y-homeomorphism is defined by a

isotopy

leaving the boundary

∂

⁡

M

1

fixed by sliding the second band

M

2

one turn around the
core of

M

1

till the original position. The result is an

automorphism

of

K

0

which maps

M

2

into itself but reversing it.

To define this for genus greater than two just consider any other non orientable surface as a connected sum of a Kein bottle plus projective planes.

1.

D.R.J. Chillingworth.

A finite set of

generators

for the
homeotopy group of a non-orientable

surface

, Proc. Camb. Phil. Soc.
65(1969), 409-430.

2.

M. Korkmaz.

Mapping Class Groups

of Non-orientable Surfaces

, Geometriae Dedicata 89 (2002), 109-133.

3.

W.B.R. Lickorish.

Homeomorphisms

of non-orientable two-manifolds

,
Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.

Title

y-homeomorphism

Canonical

name

Yhomeomorphism

Date of creation

2013-03-22 15:42:26

Last modified on

2013-03-22 15:42:26

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Definition

Classification

msc 54C10

Synonym

crosscap slide

Related topic

CrosscapSlide

Generated on Sat Feb 10 11:18:43 2018 by

LaTeXML
retract

retract

Let

X

be a

topological space

and

Y

a

subspace

of

X

. If there exists a

continuous map

r

:

X

→

Y

such that

r

⁢

(

y

)

=

y

for all

y

∈

Y

, then we say

Y

is a

retract

of

X

and

r

is a

retraction

.

Title

retract

Canonical name

Retract

Date of creation

2013-03-22 12:16:06

Last modified on

2013-03-22 12:16:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54C15

Related topic

DeformationRetraction

Related topic

PeriodOfMapping

Defines

retraction

Generated on Sat Feb 10 11:18:45 2018 by

LaTeXML
tangle

tangle

A tangle is a

1

-

manifold

, i.e. a

disjoint union

of arcs and circles, embedded in

(

0

,

1

)

2

×

[

0

,

1

]

. The

boundary

of a tangle is

contained

in

(

0

,

1

)

2

×

{

0

,

1

}

. Two tangles are considered

equivalent

if and only if they are ambient

isotopic

relative to their boundaries. Combinatorially, tangles can be understood as tangle

diagrams

. Any two tangle diagrams which

represent

the same tangle can be

connected

by

Reidemeister moves

. This is the content of a slight

generalization

of Reidemeister’s

theorem

. Algebraically, tangles form the

morphisms

of a tortile

monoidal category

. This is a corollary of Shum’s theorem. Specifically, they form the tortile monoidal category generated by a self-dual,unframed object.

Title

tangle

Canonical name

Tangle

Date of creation

2013-03-22 18:16:41

Last modified on

2013-03-22 18:16:41

Owner

apollonius (16438)

Last modified by

apollonius (16438)

Numerical id

5

Author

apollonius (16438)

Entry type

Definition

Classification

msc 54C25

Related topic

Knot

Related topic

Link

Related topic

Braid

Generated on Sat Feb 10 11:18:50 2018 by

LaTeXML
topological embedding

topological embedding

Let

X

,

Y

be

topological spaces

. A map

f

:

X

→

Y

is said to be an

embedding

(or

imbedding

) if the

restriction

f

:

X

→

f

⁢

[

X

]

is

homeomorphism

.

The notation

f

:

X

↪

Y

is often used for embeddings.

The embeddings correspond to the

subspaces

. Observe that

f

and the

inclusion map

of the subspace

f

⁢

[

X

]

into

X

differ only up to a homeomorphism.

References

1

Wikipedia’s entry on

http://en.wikipedia.org/wiki/embedding

Embedding

2

S. Willard,

General topology

, Addison-Wesley, Massachussets, 1970.

Title

topological embedding

Canonical name

TopologicalEmbedding

Date of creation

2013-03-22 15:30:59

Last modified on

2013-03-22 15:30:59

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

8

Author

kompik (10588)

Entry type

Definition

Classification

msc 54C25

Classification

msc 52B05

Synonym

imbedding

Related topic

SubspaceTopology

Generated on Sat Feb 10 11:18:52 2018 by

LaTeXML
0≠1 as real numbers

0

≠

1

as real numbers

Theorem

.

The

real numbers

0 and 1 are distinct.

There are four relatively common ways of constructing the real numbers. One can start with the

natural numbers

and augment it by adding

solutions

to particular classes of equations, ultimately considering either

equivalence classes

of

Cauchy sequences

of

rational numbers

or

Dedekind cuts

of rational numbers. One can instead define the real numbers to be the unique (up to

isomorphism

)

ordered field

with the least

upper bound

property. Finally, one can characterise the real numbers as equivalence classes of possibly

infinite

strings over the

alphabet

{

0

,

1

,

2

,

3

,

4

,

5

,

6

,

7

,

8

,

9

,

.

}

satisfying certain conditions. We offer a proof for each characterisation.

Cauchy sequences.

This construction proceeds by starting with a standard model of

Peano arithmetic

, the natural numbers

ℕ

, extending to

ℤ

by adding

additive inverses

, extending to

ℚ

by taking the

field of fractions

of

ℤ

, and finally defining

ℝ

to be the set of equivalence classes of Cauchy sequences in

ℚ

for an appropriately defined

equivalence relation

.

There is a natural embedding

i

:

ℕ

→

ℝ

defined by sending a given number

x

to the equivalence class of the

constant

sequence

(

x

,

x

,

…

)

.  Since

i

is

injective

and

0

and

1

are elements of

ℕ

, to prove that

0

≠

1

in

ℝ

we need only show that

0

≠

1

in

ℕ

.

The name

1

is a label for the

successor

S

⁢

0

of

0

in

ℕ

. One of the axioms of Peano arithmetic states that

0

is not the successor of any number. Therefore

0

≠

S

⁢

0

in

ℕ

, and so

0

≠

1

in

ℝ

.
∎

Dedekind cuts.

This construction agrees with the previous one up to constructing the rationals

ℚ

. Then

ℝ

is defined to be the set of all Dedekind cuts on

ℚ

. Letting

x

ℚ

represent the name of an element of

ℚ

and

x

ℝ

represent the name of an element of

ℝ

, we define

0

ℝ

=

{

x

∈

ℚ

|

x

<

0

ℚ

}

1

ℝ

=

{

x

∈

ℚ

|

x

<

1

ℚ

}

The proof that

0

ℚ

≠

1

ℚ

is

similar

to the previous proof. Observe that

0

ℚ

<

1

ℚ

. Since no number is less than itself, it follows that

0

ℚ

∉

0

ℝ

but

0

ℚ

∈

1

ℝ

. Thus these Dedekind cuts are not equal.
∎

Ordered field with least upper bound property.

Here the fact that

0

≠

1

is a

consequence

of the

field axiom

requiring

0

and

1

to be distinct.
∎

Decimal strings.

If one defines

0

=

(

0

,

0

,

0

,

0

,

…

)

¯

1

=

(

1

,

0

,

0

,

0

,

…

)

¯

then since neither defining string ends with a tail of 9s and the strings differ in one position, their equivalence classes are distinct.
∎

Title

0

≠

1

as real numbers

Canonical name

0ne1AsRealNumbers

Date of creation

2013-03-22 15:23:15

Last modified on

2013-03-22 15:23:15

Owner

mps (409)

Last modified by

mps (409)

Numerical id

11

Author

mps (409)

Entry type

Theorem

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

DecimalExpansion

Generated on Sat Feb 10 11:18:54 2018 by

LaTeXML
bijection between closed and open interval

bijection between closed and open interval

For

mapping

the

end points

of the closed unit interval

[

0

,

1

]

and its inner points bijectively onto the corresponding open unit interval

(

0

,

1

)

,  one has to discern suitable

denumerable

subsets in both sets:

[

0

,

1

]

=

{

0

,

1

,

1

/

2

,

1

/

3

,

1

/

4

,

…

}

∪

S

,

(

0

,

1

)

=

{

1

/

2

,

1

/

3

,

1

/

4

,

…

}

∪

S

,

where

S

:=

[

0

,

1

]

∖

{

0

,

1

,

1

/

2

,

1

/

3

,

1

/

4

,

…

}

.

Then the mapping

f

from

[

0

,

1

]

to

(

0

,

1

)

defined by

f

⁢

(

x

)

:=

{

1

/

2

for

x

=

0

,

1

/

(

n

+

2

)

for

x

=

1

/

n

(

n

=

1

,

2

,

3

,

…

)

,

x

for

x

∈

S

is apparently a

bijection

.  This means the

equicardinality

of both

intervals

.

Note that the bijection is neither

monotonic

(e.g.

0

↦

1

2

,

1

2

↦

1

4

,

1

↦

1

3

) nor

continuous

.  Generally, there does not exist any continuous

surjective

mapping

[

0

,

1

]

→

(

0

,

1

)

,  since by the

intermediate value theorem

a

continuous function

maps a

closed interval

to a closed interval.

References

1

S. Lipschutz

:

Set theory

.  Schaum Publishing Co., New York (1964).

Title

bijection between closed and open interval

Canonical name

BijectionBetweenClosedAndOpenInterval

Date of creation

2013-03-22 19:36:06

Last modified on

2013-03-22 19:36:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Example

Classification

msc 54C30

Classification

msc 26A30

Related topic

BijectionBetweenUnitIntervalAndUnitSquare

Generated on Sat Feb 10 11:19:00 2018 by

LaTeXML
index of inequalities

index of inequalities

Inequalities for reals and complex numbers

1.

The

triangle inequality

. If

a

,

b

,

c

are

real numbers

, then

|

|

a

-

c

|

-

|

b

-

c

|

|

≤

|

a

-

b

|

≤

|

a

-

c

|

+

|

b

-

c

|

.

2.

Nesbitt’s

inequality

3.

Young’s inequality (

http://planetmath.org/YoungInequality

)

4.

Bernoulli’s inequality

5.

Schur’s inequality

6.

quadratic inequality

Inequalities in an inner product space

1.

Cauchy-Schwarz-Bunyakovsky inequality (

http://planetmath.org/CauchySchwarzInequality

)

2.

Bessel inequality

3.

Peetre’s inequality

Inequalities for sequences

1.

Shapiro inequality

2.

MacLaurin’s inequality

3.

Carleman’s inequality

4.

arithmetic-geometric-harmonic means inequality

and
the

general means inequality

5.

Minkowski inequality

6.

Chebyshev’s inequality
and the

rearrangement inequality

7.

Muirhead’s

theorem

Inequalities for functions

1.

Jordan’s inequality for

sin

2.

Jensen’s inequality

3.

Gronwall’s inequality

4.

Hölder inequality

5.

Young’s inequality

6.

Wirtinger’s inequality

Geometric inequalities

1.

Hadwiger-Finsler inequality

2.

Weizenbock’s inequality

3.

Brunn-Minkowski inequality

Inequalities in number theory

1.

inequality for

fractions

(

http://planetmath.org/SummedNumeratorAndSummedDenominator

)

2.

Liouville approximation theorem

Matrix inequalities

1.

Schur’s inequality (

http://planetmath.org/ShursInequality

)

2.

Hadamard

’s inequality

3.

Wielandt-Hoffman theorem

4.

Weyl’s inequality

5.

Frobenius inequality

Inequalities for cardinal numbers

1.

Cantor’s theorem

2.

König’s theorem

Inequalities for random variables

1.

Markov’s inequality

2.

Kolmogorov’s inequality

3.

Chebyshev’s inequality (

http://planetmath.org/ChebyshevsInequality2

)

Inequalities in order theory

1.

distributive inequalities

Title

index of inequalities

Canonical

name

IndexOfInequalities

Date of creation

2013-03-22 15:41:13

Last modified on

2013-03-22 15:41:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

15

Author

matte (1858)

Entry type

Topic

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

QuadraticInequality

Generated on Sat Feb 10 11:19:02 2018 by

LaTeXML
inequalities for real numbers

inequalities for real numbers

Suppose

a

is a

real number

.

1.

If

a

<

0

then

a

is a

negative number

.

2.

If

a

>

0

then

a

is a

positive number

.

3.

If

a

≤

0

then

a

is a

non-positive number

.

4.

If

a

≥

0

then

a

is a

non-negative number

.

The first two

inequalities

are also called

strict inequalities

.

The second two inequalities are also called

loose inequalities

.

Properties

Suppose

a

and

b

are real numbers.

1.

If

a

>

b

, then

-

a

<

-

b

. If

a

<

b

, then

-

a

>

-

b

.

2.

If

a

≥

b

, then

-

a

≤

-

b

. If

a

≤

b

, then

-

a

≥

-

b

.

Lemma 1

.

0

<

a

iff

-

a

<

0

.

Proof.

If

0

<

a

, then adding

-

a

on both sides of the inequality gives

-

a

=

-

a

+

0

<

-

a

+

a

=

0

.  This process can also be reversed.
∎

Lemma 2

.

For any

a

∈

R

, either

a

=

0

or

0

<

a

2

.

Proof.

Suppose

a

≠

0

, then by

trichotomy

, we have either

0

<

a

or

a

<

0

, but not both.  If

0

<

a

,  then

0

=

0

⋅

a

<

a

⋅

a

=

a

2

.  On the other hand, if

-

(

-

a

)

=

a

<

0

, then

0

<

-

a

by the previous lemma.  Then repeating the previous ,

0

=

0

⋅

(

-

a

)

<

(

-

a

)

⁢

(

-

a

)

=

a

2

.
∎

Three direct

consequences

follow:

Corollary 1

.

0

<

1

Corollary 2

.

For any

a

∈

R

,

0

<

1

+

a

2

.

Corollary 3

.

There is no real solution for

x

in the equation

1

+

x

2

=

0

.

Inequality for a converging sequence

Suppose

a

0

,

a

1

,

…

is a

sequence

of real numbers converging to a real
number

a

.

1.

If

a

i

<

b

or

a

i

≤

b

for some real number

b

for each

i

, then

a

≤

b

.

2.

If

a

i

>

b

or

a

i

≥

b

for some real number

b

for each

i

, then

a

≥

b

.

Title

inequalities for real numbers

Canonical

name

InequalitiesForRealNumbers

Date of creation

2013-03-22 13:58:16

Last modified on

2013-03-22 13:58:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

SummedNumeratorAndSummedDenominator

Defines

strict inequality

Defines

inequality

Generated on Sat Feb 10 11:19:05 2018 by

LaTeXML