Download 1 - ckw

1. 1.2. Kinds of Texture
2.
3.
4.
5.
Optimum decomposition:
Not too many open sets: compact.
Not too few:
Hausdorff.
Probes on a topological space (S,T):
6.
External ( c.f. En ) : Homology (polygons), homotopy (loops).
7.
Internal:
Subsets of S (isolated points, sequences →
characterization).
8.
Subsets of T (rarefying coverings → texture)
9.
Consider U  S but U needs not belong to T. There exists some closed sets Cα
such that C  U .
10. Alternatively,
Closure of U  U 

C .
U   p every N  p   U    .
11. Interior of U 
intU
 U0 

O , where Oα are open sets contained in U.
12. Boundary of U  b U   U \ U 0 .
13. U 0  U \ b U  .
14. U is open → U 0  U and b U   U \ U .
15. U is closed → U  U and b U   U \ U 0 .
16. S2 embedded in E3:
S 2   S 2   S 2 → b  S 2    . S2 is both open &
0
closed.
17. A space S is connected if it has no subset with empty boundary except S & Φ.
18. p U  S is an isolated point of U if there exists N(p)  U { p }.
19. p U  S is a limit point of U if every N ' (p)  U  Φ.
20. The derived set of U  D(U)  { all limit points of U }.
21. U  U  D U  .
22. p is an accumulation (condensation) point of U if every N(p) contains countably
(uncountably) infinite points of U.
23.
24.
25.
26.
Let U,V  S. U is dense in V if U  V .
U is everywhere dense if U  S .
E.g., Q is dense in E1. Set of all rational n-tuples is dense in En.
All physical measurement deals with rational numbers (integers).
27.
28.
29.
30.
U is a nowhere dense subset if intU   , i.e., U  is everywhere dense.
The boundary of any open set is nowhere dense.
E1, seen as subset, is nowhere dense in E2, since E1' is everywhere dense.
S is separable if it has a countable everywhere dense subset, i.e., S has a
countable set P of points such that every open set in S contains at least one point
of P.
31. If S is a metric space, then S is separable is equivalent to S is 2nd separable.
32. Cantor set E 

n 0
En , where En is obtained by taking out the middle 3rd open
interval of each line segment in En–1 and E0  [0,1].
33. E is closed since it is the complement of a union of open sets.
34. At stage j, 2 j–1 intervals, each of length 1/3 j, are deleted.
35. Total length of deleted intervals 


n 1
2 j 1 1   2 
  
3j
2 n 1  3 
j
2
1 3
 
1
2 1 2
3
36. E0  Φ → E is nowhere dense.
37. There exists a 1-1 correspondence between E & [0,1].
38. Sequences are countable subsets {pn} of S.
39. A sequence {pn} converges to p , i.e., pn → p, if every open set U  p also
contains all the points in {pn | n > m } for some large enough m.
40. Let T & W be topologies on S & W is weaker than T.
41. Then every sequence that converges in T also converges in W but not vice versa.
Convergence in the stronger topology forces convergence in the weaker one.
42. On 1st countable spaces, every point in D(U) is the limit of some sequence in U.
43. Cauchy sequences {pn}:  ε >0,  k such that d  pn , pm     n, m > k.
44. Every convergent sequence is a Cauchy sequence but not vice versa.
45. If every Cauchy sequence is convergent, the metric space S is complete.
46. The set of rational numbers with induced Euclidean metric from E1 is
incomplete.
47. A sequence may converge to multiple points in a non-metric space.
48. Linear space: (V, +, α ).
49. A norm induces a norm topology on a linear space.
50. Inner product:
51. Induced norm:
V V  C
v 
52. Hilbert space H:
topology.
53. Separable version:
by
 v, u 
v, u
v, v
inner product space complete under the inner product norm
H  space with infinite but discrete basis
v
i
i  N.
54. It is isomorphic to E.
b


2
55. L2   f on  a, b with  f  x  dx    is a separable Hilbert space.


a
56. Non-separable version:
H  space with infinite continuous basis
 v
  R .
57. Addition structure (σ-algebra) required integration over α.
58. Banach space  Normed vector space complete under norm topology.
59. Finite extension as an intrinsic property ~ compactness.
60. Open covering of S  C  {Uα } such that S 

61. Heine-Borel theorem:
62.
63.
64.
65.
U .
Every open covering of I  0,1 under the topology
induced by the usual topology of E1 has a finite sub-covering.
S is a compact space if each covering of S has a finite sub-covering.
Alternative definition: If S is compact, every sequence of its points contains a
convergent sub-sequence.
Some approaches to infinite spaces:
Start with compact space with boundary, solve problem, push boundary to
infinity.
66. Periodic boundary conditions.
67. Compactify a non-compact but locally compact space.
68. Compactification by adjoining a point to the space.
69. E.g.,
identify all points at infinity to one point makes E4 → S4
70. (c.f. complex plane → Riemann sphere )
71. S is locally compact if  N(p) with compact closure  pS.
72. Every compact space is locally compact but not vice versa.
73. En is not compact but locally compact.
74. U  S is relatively compact if U is compact.
75. Riesz theorem: A normed vector space is locally compact iff its dimension is
finite.
76. Hilbert spaces are not locally compact.
77. Local compactness + separability → existence of left- & right- invariant Haar
measures.
78. S is 1st separable if given any 2 points p & q,  N(p) such that qN(p) and vice
versa.
79. S is 2nd (Hausdorff ) separable if given any 2 distinct points p & q,  N(p) & N(q)
such that N(p)N(q)  Φ.
80. Hausdorff space  2nd separable space  separated space.
81. Reminder: S is separable if it has a countable everywhere dense subset.
82. Only differential equations on Hausdorff spaces have unique solutions.
83.
Those on non-Hausdorff spaces can have only locally unique solutions.
84. The Einstein equations allows for non-Hausdorff spacetimes which leads to
causal anomalies.
85. S is a compact if it is a compact Hausdorff space.
86. A closed subspace of a compact space is compact.
87. A compact subspace is closed only if the space is Hausdorff.
88. S is normal if it is 1st countable & every 2 of its closed disjoint sets have disjoint
open neighborhoods including them.
89. Every normal space is Hausdorff but not vice versa.
90. Every metric space is normal & hence Hausdorff.
91. There are non-metric normal space.
92. E2 is not normal & non-metric.
93. Urysohn’s theorem:
S with a countable basis (S is 2nd countable) is metric iff it
is normal.
94. Ref: L.A.Steen, J.A.Seebach, “Counterexamples in Topology”, Holt (70).
95. R1 with its proper closed intervals is not a topological space (unions of closed

1 
 ,1   0,1 )
k 2  k
96. R1 with its proper half-open intervals [ p,q ) is a topological space called
intervals need not be closed:
Sorgenfrey’s line.
It is unconnected (not 2nd countable) since [ – )'  –) [–
 union of infinite
number of [ – )’s.
It is 1st countable, separable by denseness, and non-metrizable.
The Sorgenfrey topology is finer than the usual topology.
97. A covering {Uα } of S is a locally finite covering if  pS,
intersects only a finite number of Uα 's.
 N(p) that
98. A covering {Vi } is a refinement of {Uα } if  Vi  α such that Uα  Vi .
99. S is paracompact if it is Hausdorff & all its coverings have local finite
refinements.
100. A connected Hausdorff space is paracompact if it is 2nd countable.
101. Topological spaces  1st-separable  Hausdorff  Normal  paracompact 
metric.
102. Paracompactness implies
1. Partition of unity to any locally finite covering.
2. Existence of affine connection (e.g., Christoffel symbols).
103. Lorentz metric on a Hausdorff space implies paracompactness.