topology

the study of [[shapes]] and [[spaces]]
 [[caveat emptor]] my definition
Table of Contents

[20190424]
charts and atlases [[chart β homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally)]]
 [[atlas β collection of charts, covering the whole space]]
 [[if codomain of atlas is eucledian, the space is a manifold]]
[20190424]
https://en.wikipedia.org/wiki/Atlas_(topology)

[20190424]
identification of circles etc [20190123]
(2) Gluing a Sphere  YouTube [[topology]][20190123]
Union of two simply connected open subsets with pathconnected intersection is simply connected  Topospaces [[topology]] [[Data type topology]] [[topology]]

[20190126]
A Logical Interpretation of Some Bits of Topology β XORβs Hammer [[logic]]  [[Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18]]
 [[old zim notes]]
[20160618]
hausdorff spaces [[topology]]
[20190123]
(2) bothmer  YouTube [[topology]] [[viz]] [[inspiration]] [20190123]
Long line (topology)  Wikipedia[20190123]
Nsphere is simply connected for n greater than 1  Topospaces[20190126]
open set = semidecidable property [[drill]] [[topology]]
[20190424]
charts and atlases
chart β homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally)
atlas β collection of charts, covering the whole space
[20190424]
think of Earth as the space and atlas as a set of flat maps
if codomain of atlas is eucledian, the space is a manifold
local chart for manifold introduces curvilinear coordinates (coming from eucledian space)
[20190424]
https://en.wikipedia.org/wiki/Atlas_(topology)
One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
[20190424]
identification of circles etc
[20190424]
https://math.stackexchange.com/a/433969/15108 antipodal gluing of circle is circle again. nice gif
[20190424]
antipodal identification of circle (S^{1}) is { circle } [[drill]]
[20190424]
identificaiton of 2D disk: right β it's exactly the first diagram here! https://en.wikipedia.org/wiki/Real_projective_plane if you draw disk as a square.
[20190424]
this also kinda makes sense if you draw for a bit https://math.stackexchange.com/a/1391731/15108
[20190424]
antipodal identificaiton of disk (D^{2}) is { RP^{2} } [[drill]]
[20190424]
https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml
Here are classic models of the projective plane:
 The set of vectors of R^{3} with the natural topology
 A (real affine) plane completed by a projective line (line at infinity)
 A sphere where the antipodal points are identified
 A closed disk where the antipodal points of the circumference are identified
[20190123]
(2) Gluing a Sphere  YouTube [[topology]]
https://www.youtube.com/watch?v=mmkreUEoGr8
Often the fundamental group of the glued object can be calculated from the pieces (here two rectangles) and the glue (here a circle). The mathematical tool to do this is called the Seifertvan Kampen Theorem.
[20190123]
Union of two simply connected open subsets with pathconnected intersection is simply connected  Topospaces [[topology]]
Both and are trivial, so we get is an amalgamated free product of two trivial groups, hence it must be trivial.
Data type topology [[topology]]
[20190126]
Infinite compact sets
https://perl.plover.com/classes/datatopology/samples/slide022.html
onepoint compactification of β
[20190126]
Compactness
https://perl.plover.com/classes/datatopology/samples/slide021.html
Compact set = Set that can be exhaustively searched
[20190126]
Equality
https://perl.plover.com/classes/datatopology/samples/slide019.html
Discrete space = Semidecidable equality
[20190126]
Topology of Data Types
https://perl.plover.com/classes/datatopology/
[20190126]
References and further reading
https://perl.plover.com/classes/datatopology/samples/slide027.html
Other materials at http://www.cs.bham.ac.uk/~mhe/
[20190126]
A Logical Interpretation of Some Bits of Topology β XORβs Hammer [[logic]]
 State "DONE" from
[20190424]
https://xorshammer.com/2011/07/09/alogicalinterpretationofsomebitsoftopology/
[20190424]
mm, not sure how this can be useful nowβ¦
Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18
@zariskitopology So "compact" doesn't mean "small": it means "doesn't have any fuzzy edges".
<https://twitter.com/johncarlosbaez/status/1106726463607209985 >
old zim notes
[20160618]
compactness
 usual axioms of real numbers: forall a, b: a + b = b + a, forall x, y. exists z. x * z > y, so on

add constant eps
 infinite number of axioms for each n: eps < 1/n
 eps > 0
for each finite subset of eps axioms there clearly is a model with \bbR
for infinite set: no model with \bbR as domain! Nonstandard real numbers, hyperreals
[20160620]
connectedness
Connected: can't be represented as a union of two disjoint open sets.
Locally connected at x: for every open V(x), there is connected open U(x) β V(x). X is locally connected if locally connected at every point.
Local connectedness and connectedness are unrelated!
Path connected: there is a path joining every pair of points.
Locally path connected at x: for every open V(x), there is connected open U(x) \subseteq V(x). X is locally path connected if locally path connected at every point.
Simply connected: pathconnected and fundamental group is trivial.
Locally simply connected: admits a base of simply connected sets. Also locally pathconnected and locally connected.
[20150614]
Extracting topology from convergence
f_{n} > weak(*) f if forall x. f_{n}(x) > f(x)
How to develop intuition abut the open sets?
f_{n} converges weakly to f if it converges pointwise
f_{n} converges weakly to f:
forall O(f). exists N. forall n > N. f_{n} β O
What is O? finite number of points do not converge?
[20160618]
hausdorff spaces [[topology]]
Hausdorff if any two points can be separated by neighborhoods (diagonal is closed in product topology).
Space X is Hausdorff iff its apartness map
β : X x X > S
(x, y) > { x β y }
is continuous
Space is discrete if every singleton is open (or if its diagonal is open)
Space is discrete iff its equality map
\eq : X x X > S
(x, y) > { x = y }
is continuous
[20190123]
(2) bothmer  YouTube [[topology]] [[viz]] [[inspiration]]
https://www.youtube.com/channel/UCngLGVygGfVo3pxsRzeCN_A
[20190224]
some topology visualisations
[20190123]
Long line (topology)  Wikipedia
https://en.wikipedia.org/wiki/Long_line_(topology)
[20190123]
Nsphere is simply connected for n greater than 1  Topospaces
https://topospaces.subwiki.org/wiki/Nsphere_is_simply_connected_for_n_greater_than_1
[20190126]
open set = semidecidable property [[drill]] [[topology]]
 public document at doc.anagora.org/topology
 video call at meet.jit.si/topology
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