# General topology

## Table of Contents

- mathtopology
charts and atlases
- chart – homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally) mathtopology
- atlas – collection of charts, covering the whole space mathtopology
- mathtopology 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 mathtopology
- mathtopology https://en.wikipedia.org/wiki/Atlas_(topology)

- mathtopology
identification of circles etc
- mathtopology
https://math.stackexchange.com/a/433969/15108 antipodal gluing of circle is circle again. nice gif
^{1}) is { circle } mathtopologydrill antipodal identification of circle (S

- mathtopology
identificaiton of 2D disk: right – it's exactly the first diagram here! https://en.wikipedia.org/wiki/Real_projective_plane if you draw disk as a square.
- mathtopology this also kinda makes sense if you draw for a bit https://math.stackexchange.com/a/1391731/15108
^{2}) is { RP^{2}} mathtopologydrill antipodal identificaiton of disk (D- mathtopology https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml

- mathtopology
https://math.stackexchange.com/a/433969/15108 antipodal gluing of circle is circle again. nice gif
- [C] (2) Gluing a Sphere - YouTube mathtopology
- [C] Union of two simply connected open subsets with path-connected intersection is simply connected - Topospaces mathtopology
- [D] Data type topology mathtopology
- DONE [C] A Logical Interpretation of Some Bits of Topology – XOR’s Hammer mathtopologylogic
- mathtopologylogic mm, not sure how this can be useful now…

- TODO [D] Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18 mathtopology
- TODO [C] old zim notes mathtopology
- mathtopology compactness
- mathtopology connectedness
- mathtopology Extracting topology from convergence

- mathtopology hausdorff spaces
- [C] (2) bothmer - YouTube mathtopologyvizinspiration
- mathtopologyvizinspiration some topology visualisations

- [D] Long line (topology) - Wikipedia mathtopology
- [D] N-sphere is simply connected for n greater than 1 - Topospaces mathtopology
- mathtopologydrill open set = semidecidable property

## ¶ charts and atlases mathtopology

### ¶chart – homeomorphism from an open subset of manifold to some other space (not necessarily eucledian generally) mathtopology

### ¶atlas – collection of charts, covering the whole space mathtopology

#### ¶ think of Earth as the space and atlas as a set of flat maps mathtopology

### ¶if codomain of atlas is eucledian, the space is a manifold mathtopology

#### ¶local chart for manifold introduces curvilinear coordinates (coming from eucledian space) mathtopology

### ¶ https://en.wikipedia.org/wiki/Atlas_(topology) mathtopology

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.

## ¶ identification of circles etc mathtopology

### ¶ https://math.stackexchange.com/a/433969/15108 antipodal gluing of circle is circle again. nice gif mathtopology

#### ¶ antipodal identification of circle (S^{1}) is { circle } mathtopologydrill

### ¶ identificaiton of 2D disk: right – it's exactly the first diagram here! https://en.wikipedia.org/wiki/Real_projective_plane if you draw disk as a square. mathtopology

#### ¶ this also kinda makes sense if you draw for a bit https://math.stackexchange.com/a/1391731/15108 mathtopology

#### ¶ antipodal identificaiton of disk (D^{2}) is { RP^{2} } mathtopologydrill

#### ¶ https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml mathtopology

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

## ¶[C] (2) Gluing a Sphere - YouTube mathtopology

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 Seifert-van Kampen Theorem.

## ¶[C] Union of two simply connected open subsets with path-connected intersection is simply connected - Topospaces mathtopology

Both and are trivial, so we get is an amalgamated free product of two trivial groups, hence it must be trivial.

## ¶[D] Data type topology mathtopology

### ¶[C] Infinite compact sets mathtopology

https://perl.plover.com/classes/data-topology/samples/slide022.html

one-point compactification of ℕ

### ¶[C] Compactness mathtopology

https://perl.plover.com/classes/data-topology/samples/slide021.html

Compact set = Set that can be exhaustively searched

### ¶[C] Equality mathtopology

https://perl.plover.com/classes/data-topology/samples/slide019.html

Discrete space = Semidecidable equality

### ¶[C] Topology of Data Types mathtopology

### ¶[C] References and further reading mathtopology

https://perl.plover.com/classes/data-topology/samples/slide027.html

Other materials at http://www.cs.bham.ac.uk/~mhe/

## ¶DONE [C] A Logical Interpretation of Some Bits of Topology – XOR’s Hammer mathtopologylogic

- State "DONE" from

https://xorshammer.com/2011/07/09/a-logical-interpretation-of-some-bits-of-topology/

### ¶ mm, not sure how this can be useful now… mathtopologylogic

## ¶TODO [D] Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18 mathtopology

@zariskitopology So "compact" doesn't mean "small": it means "doesn't have any fuzzy edges".

https://twitter.com/johncarlosbaez/status/1106726463607209985

## ¶TODO [C] old zim notes mathtopology

### ¶ compactness mathtopology

- 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

- infinite number of axioms for each n: eps < 1/n

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

### ¶ connectedness mathtopology

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) ⊂ 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: path-connected and fundamental group is trivial.

Locally simply connected: admits a base of simply connected sets. Also locally path-connected and locally connected.

### ¶ Extracting topology from convergence mathtopology

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} ∈ O

What is O? finite number of points do not converge?

## ¶ hausdorff spaces mathtopology

Hausdorff if any two points can be separated by neighborhoods (diagonal is closed in product topology).

Space X is Hausdorff iff its apartness map

≠ : X x X -> S

(x, y) -> { x ≠ 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