Ubuntu Mobile
Ubuntu Mobile’s finally released!
In personal news, my Topology professor is challenging me to create a more efficient Python program that deals with this problem from our Topology homework:
The set consisting of two elements {a, b} can be given four different topologies:
1. {∅, {a, b}} – The indiscrete topology
2. {∅, {a}, {b}, {a, b}} – The discrete topology
3. {∅, {a}, {a, b}}
4. {∅, {b}, {a, b}}
Each of these is a topology on {a, b} because each one includes {a, b} and is closed
under ∩ and . There are no other topologies on {a, b} because any other set of subsets
would fail to include ∅ or {a, b}.
Determine all of the topologies on the three element set {a, b, c}. Remember that a
topology on X is a set of subsets of X. There are 23 subsets of X. So there are at 3
most 22 = 256 possibilities, most of which fail to be topologies. Any topology must
include ∅ and {a, b, c}. So this cuts down the search. In particular, since we have
no choice but to include these two subsets, the actual subsets of {a, b, c} that are
“up for grabs” are only the remaining 23 − 2 of them. That is, you only need to check 3
22 −2 = 64 possibilities. Even for these, many will fail to be topologies. For example,
τ = {∅, {a}, {b}, {c}, {a, b, c}} is not a topology because {a} ∈ τ and {b} ∈ τ but
{a} ∪ {b} ∈/ τ.
Set theory… brutal.
February 27th, 2008 at 9:45 pm
…what does that even mean?!