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# test

1 FtestF2010/03/20(y) 07:28:12
test

2 FtestF2010/03/20(y) 07:29:53
ƉF̖{̖܂B

3 FPRQlڂ̑fF2010/03/20(y) 10:04:52
˃NY

4 FtestF2010/03/20(y) 20:29:57
>3 ʕ񎀃}X^^

5 FPRQlڂ̑fF2010/03/20(y) 21:25:05
http://at.yorku.ca/list/qa.htm#ta

6 FPRQlڂ̑fF2010/03/20(y) 21:55:54
ctx炵񂾂Aꕪ邩?@ɂ͕񂩂B

u66233̂A234432999̏ꍇA56890̂22310͂ɂȂ邩Hv

7 FPRQlڂ̑fF2010/03/21() 01:21:47
When does the 0-dimensional Reflection Preserve Products?

Andrej Bauer

We say that a topological space is zero-dimensional when the collection of its clopen (closed and open) subsets forms a basis for its topology.

For a topological space X, let z(X) be its zero-dimensional reflection, i.e., the same underlying set X but with the topology generated by the collection of the clopen subsets of X.

Question 1: Does z preserve products, i.e., is z(X ~Y) homeomorphic to z(X) ~z(Y)?

The question can be stated equivalently as follows.

Question 1': Is every clopen subset of X ~Y a union of the sets of the form U ~V, where U 'subset or equal' X is clopen and V 'subset or equal' Y is clopen?

8 FPRQlڂ̑fF2010/03/21() 01:32:14
The Sierpinski Triangle

Sonia Sabogal

The Sierpinski Triangle is obtained as the residual set remaining when
one begins with a triangle and applies the operations of dividing it
into four triangles, connecting the middle points of the triangle's
edges, and omitting the interior of the center one, then repeats this
operation on each of the surviving 3 triangles, then repeats again on
the surviving 9 triangles, and so on indefinitely. This space is
homeomorphic to the unique nonempty compact set K of the complex plane
that satisfies: K = w1(K) \cup w2(K) \cup w3(K) where w1, w2, w3 are
maps of the complex plane defined by: w1(z) = 0.5z, w2(z) = 0.5z+0.5
and w3(z) = 0.5z+0.5i (see W. J. Charatonik and A. Dilks; Topol. and
its Appl.; 55(1994), 215-238, Example 2.7).

Question: Is there a known topological characterization of Sierpinski triangle?

9 FPRQlڂ̑fF2010/03/21() 02:44:23
>>7
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10 FPRQlڂ̑fF2010/03/21() 12:51:26
Theorem 4.1.1: For a topological space (X, T), the following are true:
(i) and X are closed sets
(ii) Intersection of any number of closed sets is closed
(iii) Union of finitely many closed sets is closed.

Ӂ(X, T)[^c=closed
AUBU...<(X,T)->(AUBU...)^c=A^c&B^c&..=closed
A&B&...<(X,T)->(A&B&...)^c=A^cUB^cU..=closed

11 FPRQlڂ̑fF2010/03/21() 13:04:33
Theorem 4.2.1: In a topological space the following are euqivalent:
(i) G is open
(ii) G = G0

G<(X,T)->G<G<(X,T)->G<G0
G0<G
so G=G0

12 FPRQlڂ̑fF2010/03/21() 13:15:30
Theorem 4.2.2: In a topological space (X, T), the following are equivalent:
(i) K0 = U {G < K; G<T}
(ii) K0 is the largest open set contained in K.
(i)->K0<G<K->K0<K0,K0^c<!G<T
(i)->K0=UG<T->K0=open
G<S<K,S<T->S<U {G < K; G<T}=K0->S<K0

13 FPRQlڂ̑fF2010/03/22() 07:16:19
Theorem 4.2.4: The following are equivalent in a space (X, T)
(i) K~ = & {F > K; F is closed in X}
(ii) K~ is smallest closed set containing K.

(i)->F=F~=F+F'->K~=K+K'<! F->K'(>a)<!F->Na'&K=p<F=F+F'->Na'&F=!->a<F'!!
K~<F->b<!K~<F->K~=closed<F->K~(>K)&(F>K)<&(F>K)->b<K~!!->(ii)

14 FPRQlڂ̑fF2010/03/22() 10:20:57
Theorem 4.2.5: In (X, T) the following are equivalent
(i) F is closed
(ii) F = F~=F+F'
(iii) F' < F

(i)->F=F+F'>F'->(iii)->(i)
(ii)->F'<F->(iii),(iii)->F+F'<F<F+F'->(ii)
(i)=(iii)=(ii)

F=F&F'

15 FPRQlڂ̑fF2010/03/22() 10:59:34
Theorem 4.2.6: In a topological space (X, T), let Np denote the family of all neighbourhood of a
point p of X, then the following are true
(i) p belongs to each member of Np .
(ii) Every superset of a member of Np belongs to Np .
(iii) For every member N of Np there is another member G of Np , which is the neighbourhood
of each of its points.

(i)p<G<Np
(ii)V,U<Np->(V,U)<Np=UNp
(iii)N<Np->p<G<N<Np,G=open->G<Np and G<Ng,g<G

16 FPRQlڂ̑fF2010/03/22() 16:13:43
Theorem 4.4.1: A non-empty class of subsets of a set X is a base for a topology of X iff
(i) X = {B; B<B}
(ii) For any B1,B2<B B1&B2 is the union of members of B, i.e., if p <B1&B2, then there
exists B'<B such that p<B'<B<B1&B2.

(i)->,X<UB
(ii)->&T<(UB)&(UB)=U(B&B)=UB'

17 FPRQlڂ̑fF2010/03/22() 18:36:31
First Axiom of Countability. A topological space has a countable local base at each of its points.
Second Axiom of Countability. A topological space has a countable base.

Definition: A topological space is called first countable (respectively second countable) if it satisfies
the first (respectively second) axiom of countability.

Lindelofffs Theorem: Every non-empty open set in a second countable space can be represented by a
countable union of basic open sets.

Corollary: Every open base of a second countable space has a countable subclass which is also an
open base.

Theorem 4.4.2: Every second countable space is separable.

18 FPRQlڂ̑fF2010/03/22() 19:09:59
Theorem 4.4.3: Every separable metric space is second countable.

19 FPRQlڂ̑fF2010/03/22() 19:42:13
Theorem 5.1.1: A topological space (X, T) is a T1 -space iff every singleton subset of X is closed.

x,y<X->x^c=open>y,y^c=open>x

20 FPRQlڂ̑fF2010/03/22() 19:44:59
Corollary: Every finite subset of a T1 -space is closed.

S=U{x}=Closed

21 FPRQlڂ̑fF2010/03/22() 19:51:00
Definition: A topological space (X, T ) is a called a T2 -space or a Hausdorff space if it satisfies the T2 -
axiom of separation.
Note the following then:
(a) Every Hausdorff space is a T1-space.
(b) Every metric space is a Hausdorff space.
(c) Every subspace of Hausdorff space is Hausdorff.
(d) The cartesian product of Hausdorff spaces is Hausdorff.
(e) The set {(x, x); x < X} is closed in X x X if X is Hausdorff.
(f) Every convergent sequence in a Hausdorff space converges to a unique limit.

22 F 27Tn7FHaVY F2010/03/23() 00:28:06

23 FghclfYsc82 F2010/03/23() 08:44:55
Mathematics is very good to be discussed here at 2chan.

--neko--

24 FPRQlڂ̑fF2010/03/25() 21:00:46
conquer the universe

Aimy high/

Behind the Old French expression that evolved into Amy is a more ancient Latin verb, amare, meaning ''to love.''

25 FPRQlڂ̑fF2010/05/06() 07:44:26
test

26 FPRQlڂ̑fF2010/06/22() 15:25:55
test

27 FPRQlڂ̑fF2010/06/23() 20:31:53
test

28 FPRQlڂ̑fF2010/06/24() 20:43:15

29 F p177219.amixcom.jpF2010/06/25() 03:41:53
tes

30 FfusianasanF2010/06/25() 03:43:06
tes

31 FfusianasanF2010/06/25() 03:44:51
test

32 FPRQlڂ̑fF2010/06/28() 21:26:24
test

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