X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

Question

I'm teaching my self topology with the aid of a book. I'm trying to do the following problem:

Let X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

I'm not sure If I totally understand what is being asked, but I'm going to attempt to list every topology in groups that are homeomorphic to one another.

I want to know if this is correct.

(A) trivial topology. $\mathscr{T}=${X,$\varnothing$}; I can't think of anything else that is homeomorphic to this one.

(B) "singles"
(B1): $\mathscr{T}=${X,$\varnothing$,{1}};
(B2): $\mathscr{T}=${X,$\varnothing$,{2}};
(B3): $\mathscr{T}=${X,$\varnothing$,{3}};

(C) "doubles"
(C1): $\mathscr{T}=${X,$\varnothing$,{1,2}};
(C2): $\mathscr{T}=${X,$\varnothing$,{2,3}};
(C3): $\mathscr{T}=${X,$\varnothing$,{3,1}};

(D) "single-doubles"
(D1): $\mathscr{T}=${X,$\varnothing$,{1},{1,2}};
(D2): $\mathscr{T}=${X,$\varnothing$,{1},{1,3}};
(D3): $\mathscr{T}=${X,$\varnothing$,{2},{2,1}};
(D4): $\mathscr{T}=${X,$\varnothing$,{2},{2,3}};
(D5): $\mathscr{T}=${X,$\varnothing$,{3},{3,1}};
(D6): $\mathscr{T}=${X,$\varnothing$,{3},{3,2}};

(D') "single-doubles (disjoint)"
(D'1): $\mathscr{T}=${X,$\varnothing$,{3},{1,2}};
(D'2): $\mathscr{T}=${X,$\varnothing$,{2},{1,3}};
(D'3): $\mathscr{T}=${X,$\varnothing$,{1},{2,3}};

(E) "single-single-doubles"
(E1): $\mathscr{T}=${X,$\varnothing$,{1},{2},{1,2}};
(E2): $\mathscr{T}=${X,$\varnothing$,{1},{3},{1,3}};
(E3): $\mathscr{T}=${X,$\varnothing$,{2},{3},{2,3}};

(F) "single-double-doubles"
(F1): $\mathscr{T}=${X,$\varnothing$,{1},{1,2},{1,3}};
(F2): $\mathscr{T}=${X,$\varnothing$,{2},{2,1},{3,2}};
(F3): $\mathscr{T}=${X,$\varnothing$,{3},{3,2},{3,1}};

(G) "single-single-double-doubles"
(G1): $\mathscr{T}=${X,$\varnothing$,{1},{2},{1,2},{2,3}};
(G2): $\mathscr{T}=${X,$\varnothing$,{1},{2},{1,2},{3,1}};
(G3): $\mathscr{T}=${X,$\varnothing$,{1},{3},{1,2},{3,1}};
(G4): $\mathscr{T}=${X,$\varnothing$,{1},{3},{2,3},{3,1}};
(G5): $\mathscr{T}=${X,$\varnothing$,{2},{3},{2,3},{3,1}};
(G6): $\mathscr{T}=${X,$\varnothing$,{2},{3},{1,2},{2,3}};

(H) power set: $\mathscr{T}=${X,$\varnothing$,{1}, {2},{3},{1,2},{2,3},{3,1}};; I can't think of anything else that is homeomorphic to this one.

IS this a complete list of all topologies on X?

score 6 · Accepted Answer · answered Jun 24 '16 at 20:38

6

You’re missing the ones homeomorphic to $\big\{\varnothing,X,\{1\},\{2,3\}\big\}$; there are $3$ of those. Also, your (E) group lists one topology twice: (E1) and (E3) are the same. The question wants you to list one topology from each of the $9$ groups (including the group that I just added).

answered Jun 24 '16 at 20:38

Brian M. Scott

616,228

Thanks, I'll fix it later. E1 and E3 is just cause i didn't double check. But i didn't even think of {1},{2,3}. – Michael Maliszesky Jun 24 '16 at 22:30
Is {∅,X,{1},{2,3}}{∅,X,{1},{2,3}} homeomorphic with {∅,X,{2},{2,3}}{∅,X,{1},{2,3}} ? – Michael Maliszesky Jun 25 '16 at 14:05
@Michael: $\big{\varnothing,X,{1},{2,3}\big}$ is homeomorphic to $\big{\varnothing,X,{2},{1,3}\big}$ and to $\big{\varnothing,X,{3},{1,2}\big}$. – Brian M. Scott Jun 25 '16 at 18:51
So my D and D' groups are not homeomorphic to one another? – Michael Maliszesky Jun 26 '16 at 03:39
@Michael: No, they're not: the D' topologies contain a pair of non-empty open sets, and the D topologies do not. The D spaces are connected; the D' spaces are not. – Brian M. Scott Jun 26 '16 at 08:59

X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

1 Answers1