Immersion (mathematics) - Wikipedia

Immersion (mathematics) - Wikipedia: "For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in R2), though it embeds in codimension 1 (in R3).

"



'via Blog this'

Orientability - Wikipedia

Orientability - Wikipedia: " a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to a loop going around the opposite way."



 that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability.

'via Blog this'

Topological group - Wikipedia

Topological group - Wikipedia: "The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous."



'via Blog this'

An Approach to Greek Lettering

by Michael A. Covington



'via Blog this'

Greek alphabet names in greek

barbershop is µπαρµπερικo ´ = barberiko



'via Blog this'

[PDF] Schuller's Geometric Anatomy of Theoretical Physics, Lectures 1-17 - Free Download PDF

[PDF] Schuller's Geometric Anatomy of Theoretical Physics, Lectures 1-17 - Free Download PDF:

 {b} is neither open nor closed

Example 5.7. The interval [0, 1] is compact in (R, Ostd). The one-element set containing (−1, 2) is a cover of [0, 1], but it is also a finite subcover and hence [0, 1] is compact from the definition.

It is clear that removing even one element from C will cause C to fail to be an open cover of R. Therefore, there is no finite subcover of C and hence, R is not compact.

Example

4.6

.

Let

M

=

{

a,b,c

}

and let

O

=

{

∅

,

{

a

}

,

{

a,b

}

,

{

a,b,c

}}

. Then

{

a

}

is open

but not closed,

{

b,c

}

is closed but not open, and

{

b

}

is neither open nor close

Example

4.6

.

Let

M

=

{

a,b,c

}

and let

O

=

{

∅

,

{

a

}

,

{

a,b

}

,

{

a,b,c

}}

. Then

{

a

}

is open

but not closed,

{

b,c

}

is closed but not open, and

{

b

}

is neither open nor close

'via Blog this'

Incidence geometry - Wikipedia

Incidence geometry - Wikipedia: "Every triple of distinct points is incident with precisely one cycle.
For any flag (P, z) and any point Q not incident with z there is a unique cycle z∗ with P I z∗, Q I z∗ and z ∩ z∗ = {P}. (The cycles are said to touch at P.)
Every cycle has at least three points and there exists at least one cycle."



'via Blog this'

Homogeneous space - Wikipedia

Homogeneous space - Wikipedia: " Erlangen program, one may understand that "all points are the same", in the geometry of X. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenth century.

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups. "



'via Blog this'

Erlangen program - Wikipedia

Erlangen program - Wikipedia: "only the incidence structure and the cross-ratio are preserved under the most general projective transformations."



'via Blog this'

Why is group theory important?

Why is group theory important?: "we expect the laws of physics to be unchanging in time. This is an invariance under "translation" in time, and it leads to the conservation of energy. Physical laws also should not depend on where you are in the universe. Such invariance of physical laws under "translation" in space leads to conservation of momentum. Invariance of physical laws under (suitable) rotations leads to conservation of angular momentum. "



'via Blog this'