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## Main Question or Discussion Point

I was wondering about topology.

a) Is there an algorithm for the number of topologies on finite sets?

b) If two spaces are homeomorphic, are intersections of opens sent to intersections of opens? Are unions of opens sent to unions of opens?

I tried to find an algorithm in the first part, and tried to prove the second part, both cases I failed. Anyone knows if there is a way to do this? The lecture notes I use does not seem to speak of it.

For the first one, I convinced myself that if it exists, it was very hard to do... so it probably doesn't exist or isn't found yet, otherwise I'd expect my notes to mention it.

For the second one, I suspect it's very easy (I'm just reading the chapter on homeomorphisms) because homeomorphisms are called 'the isomorphisms of topology'

a) Is there an algorithm for the number of topologies on finite sets?

b) If two spaces are homeomorphic, are intersections of opens sent to intersections of opens? Are unions of opens sent to unions of opens?

I tried to find an algorithm in the first part, and tried to prove the second part, both cases I failed. Anyone knows if there is a way to do this? The lecture notes I use does not seem to speak of it.

For the first one, I convinced myself that if it exists, it was very hard to do... so it probably doesn't exist or isn't found yet, otherwise I'd expect my notes to mention it.

For the second one, I suspect it's very easy (I'm just reading the chapter on homeomorphisms) because homeomorphisms are called 'the isomorphisms of topology'

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