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1.1. Topological Spaces
1.
Physical quantities: every acceptable value has a neighorhood of accetable values
→ open sets → topology → continuity, convergence, ….
2.
3.
Space  point set with structure (organization between its parts).
Issues: definition of neighborhood, distinguishability of points, continuity,
differentiablity.
4.
Topology of a point set S  Family T of subsets (open sets) of S such that
a. S & Φ  T.
U j T
b.
finite j
U j T
c.
j
5.
6.
7.
8.
Discrete topology  all subsets of S.
Indiscrete topolgy  { S, Φ }.
Topological space (S,T)  point set S with topology T.
Euclidean space  Rn with the usual topology (open-balls).
9.
Finite space:
T   a, a, b, b, c, d , S,   .
S  a, b, c, d  ,
10. The neighborhood of p  S is any set U such that W  U , p W and W is
an open set.
11. Euclidean space En as a topological space:
Point set  p   p1 ,
, pn 
Distance function  d  p, q  
 p
n
i 1
i
 qi 
2
Open set  open balls
12. Topology basis  B   U

such that for all open set V  T and all points
p V , there exists some U  B such that p U  V .
13. First-countable topological space S:
neighhoods
N 
j
every point p in S has a countable set of
such that  U  p , there exists at least one N j  U .
14. B & B’ define the same topology iff  U  B and p U , there exists some
U   B such that
p  U   U  and vice versa.
15. Second-countable topological space S: S has at least one countable basis.
16. All 2nd countables are 1st countables, but not necessarily the reverse.
17. Induced topology: Let (S,T) be a topological space and XS. The induced
(relative) topology is defined as XT. This gives rise to an induced topological
space (X, XT).
18. The upper-half space En   R n , R n  T  , where T is the usual topology of En ,

R n  p   p1 ,
, pn   Rn ; pn  0

19. En is 1st-countable but not 2nd-countable.
( Open balls tangent to the
hyperplane p n  0 must include the tangent point.
These balls are
uncountable. )
20. Let (S,T1) & (S,T2) be 2 topological spaces.
T1 is weaker than T2 if every member of T1 belongs to T2.
T1 is then coarser than T2 & T2 is finer (stronger) than T1.
21. Topology of Minkowski space is not known. One choice is the Zeeman
topology: the finest topology on R4 which induces an E3 topology on space
sector & E1 topology on time-axis. It’s not 1st countable.
22. The complement of an open set is closed.
23. S & Φ are both open & closed.
24. A connected space is a topological space in which no proper subset is both open
& closed → S can’t be decomposed into 2 disjointed open sets.
25. The set of all subsets of S is the power set P(S) or 2S.
26. Discrete topology: T  2S. All sets are both open & closed. E.g., topology
induced on the light cone by the Zeeman topology. Since no mapping fron E1
to a discrete space can be continuous, a photon hops from point to point on the
light-cone.
1
d  p, q   
0
28. A prametric is is a mapping:
27. Discrete metric:
 : S  S  R
such that
if
pq
pq
  p, p   0
 pS
29. A prametric is sufficient to define a topology.
30. Topological (cartesian) product of (A,T’) & (B,T”) with bases U & V, resp, is
(AB, T’T”) with basis UV.
31. The n-torus Tn is the cartesian product of n S1.
32. Convergence: filters.
Integration: Borel σ-algebra.