The following is taken from wikipedia: http://en.wikipedia.org/wiki/Finite_topological_space
2 points
Let $X = \{a,b\}$ be a set with 2 elements.
There are four distinct topologies on $X$:
- $T_1$: $\{\emptyset, \{a,b\}\}$ (the trivial topology)
- $T_2$: $\{\emptyset, \{a\}, \{a,b\}\}$
- $T_3$: $\{\emptyset, \{b\}, \{a,b\}\}$
- $T_4$: $\{\emptyset, \{a\}, \{b\}, \{a,b\}\}$ (the discrete topology)
The second and third topologies above are easily seen to be homeomorphic. The function from $X$ to itself which swaps $a$ and $b$ is a homeomorphism. A topological space homeomorphic to one of these is called a Sierpiński space. So, in fact, there are only three inequivalent topologies on a two point set: the trivial one, the discrete one, and the Sierpiński topology. The specialization preorder on the Sierpiński space $\{a,b\}$ with $\{b\}$ open is given by: $a \le a$, $b \le b$, and $a \le b$.
I am looking for homeomorphic topologies apart from this "easily seen" one $(X,T_2) \to (X,T_3)$.
These are bijections that are continuous with continuous inverses.
Is $(X,T_1) \to (X,T_2)$
and
$(X,T_2) \to (X,T_1)$
homeomorphic?Are there any others?
Best Answer
No there aren't any others. (If I understand your question correctly.) How many bijections can you have on $X$? Two, namely $f_1(x) = id_X$ and $f_2: a \mapsto b, b \mapsto a$.
Why is neither of these a homeomorphism between $T_1$ and $T_2$? Because $f^{-1}(\{a\}) = \{a\}$ is not open in $T_1$ hence $f_1$ (and similarly $f_2$) is not continuous in this case.
What about $T_2 \to T_1$? Well, a homeomorphism has to be open but $f_1(\{a\}) = \{a\}$ is not open. So again, neither of the $f_i$ is a homeomorphism.