What's the most dastardly space to work in?

No. What about that topology for Z that can be used to proves there are infinitely many primes. That is not a metric topology.

Wow, so useful...
Like infinitely many primes cannot be proved in a thousand other ways.

The function spaces C(X,Y) can be given various topologies which are not metrizable (unless X and Y are very really nice spaces), these are very useful

kek

>what is the zariski topology

you can do this with any partial order. helps you get better intuition for these things, and such points do show up elsewhere (generic points).

Non-hausdorff spaces are dastardly af (also people saying things like banach/hilbert spaces can't know what dastardly means). Pretty much anything you can imagine happens as you think it would in banach/hilbert spaces.

Sobolev Spaces, they are quite unwieldy things.

Define useful.

coming more from logic and algebra, being sober (and so T0) is plenty for me.

this made me question my intuition of topology

you seem to associate some abstract notion of distance to a and b, but i'm not sure i see it

what does the sierpinski topology "look like" to you

It doesn't look like anything. Even though it generalizes metric spaces, topology is a very abstract concept and a general topological spaces behaves nothing like a metric space. That's precisely why we place further restrictions like Hausdorff condiction, first countability, some local connectedness etc. on our spaces, it's to make them behave more like the familiar metric spaces.

Something like Formal Schemes. You have to deal with the standard topology of the scheme, but also the adic topologies of the rings.

You can approach a but you can't approach b.

brainlet tier b8

>Even though it generalizes metric spaces
That is a really fucked up way to look at topology, user.

>It doesn't look like anything.
Of course you would reach this conclusion if you're thinking of topology as a generalization of metric spaces. Protip: It is not that at all. Construct some very simple topologies over sets with a few elements (less than 10) and start building some intuition. As an exercise maybe pick a set and then compare a bunch of different topologies on it and see how they're related with each other. Then start playing with countably infinite sets and uncountably infinite sets and really start seeing what limits (of all different types) are and what it means for a space to be compact (according to all the different definitions of compactness).

>sierpinski topology,
think of it as the frame of the locale S, such a frame is really the scott topology on the ideal completion of the poset 2

Get a load of this brainlet.

By the way, my name is Tai-Danae Bradley. (It's nice to meet you!) I'm a third year math student in the PhD program at the CUNY Graduate Center, and my research interests lie in algebraic topology with applications to quantum physics.

math3ma.com/mathema/2016/10/6/the-sierpinski-space-and-its-special-property

This is an intuitive explanation of the major properties of the Sierpinski topology.

I looked through your site a bit and it looks like you do a good amount of category theory with mathematics. Maybe you can help me with a problem I've been having. I know category theory and I've been trying to learn Algebra through category theory. Ordinarily people do this the other way around and unfortunately all the textbooks I've found talking about algebra and category tend to have handwave big issues like what category they're in or what their objects are. My main problems are definitions, in particular I'm totally stumped with regards to group actions right now.

A group action from a group to a set seems simple, just a functor from the group's category (single object category w/arrows as elements) to Set (where arrows map to automorphisms in Set).

A group action from a group to itself seems like it ought to be a functor from the group's category to the underlying set of a group (with arrows mapping to automorphisms).

A group acting on itself via conjugation is totally fucked up and I have absolutely no idea how to implement it without bullshitting the reader and hoping they won't notice.

Cunny, you say?

>CUNY Graduate Center
hahahahahahahahahahahahahahaahhaa
cunny

>what does the sierpinski topology "look like" to you
A special point and a generic point.

...

wtf is this meme and the ordering is all wrong

omg, it feels like everyone is studying or trying to do "outreach" about operads rn

>when women make memes

I am getting an irrational urge to not let this meme vote and treat it as property.

ok i poke fun but your blog would have been useful to me a few years ago

I'm finding it useful. Definitely thankful for this sort of stuff being out there, especially the drawings. That shit looks easy but it takes dedication to do.

I wish there more blogs like it during my thesis. Kids have it easier this days.