General topology
Table of Contents
- 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
- if codomain of atlas is eucledian, the space is a manifold mathtopology
- https://en.wikipedia.org/wiki/Atlas_(topology) mathtopology
- identification of circles etc mathtopology
- [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
- TODO [D] Tweet from John Carlos Baez (@johncarlosbaez), at Mar 16, 01:18 mathtopology
- TODO [C] old zim notes mathtopology
- hausdorff spaces mathtopology
- [C] (2) bothmer - YouTube mathtopologyvizinspiration
- [D] Long line (topology) - Wikipedia mathtopology
- [D] N-sphere is simply connected for n greater than 1 - Topospaces mathtopology
- open set = semidecidable property mathtopologydrill
¶ 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 (S1) 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 (D2) is { RP2 } mathtopologydrill
¶ https://mathcurve.com/surfaces.gb/planprojectif/planprojectif.shtml mathtopology
Here are classic models of the projective plane:
- The set of vectors of R3 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
fn -> weak(*) f if forall x. fn(x) -> f(x)
How to develop intuition abut the open sets?
fn converges weakly to f if it converges pointwise
fn converges weakly to f:
forall O(f). exists N. forall n > N. fn ∈ 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