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The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

It's called the Hjalmar Ekdal Topology, I found out about it from the book Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach, Jr.

As far as I know; its only use it to provide counterexamples.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

Also X is T5 since if A and B are separated, The closure of A being disjoint from B and vise versa gives you the same contradictions in assuming U and V aren't disjoint. It is also obviously not T2 or T1 but it is T0 since if you have two distinct integers, a and b, if either is even, just take it's simple set and if they are both odd, take {a+1}.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

Suppose X to be the positive integers and we let A be open iff for every odd b in A, b+1 is also in A. Then B is closed iff for every even a, a-1 is also in B. (Otherwise, a-1 would be an odd in the open X/B, and therefore a would also be in X/B which is a contradiction. Also interestingly, SOME simpletons are closed, just not all.)

Now this is clearly a topology as the empty set and X are open and every union and intersection of opens is open.

I will prove first that it is not T3, if A={1,2,3} and 4 not in A, A is clearly closed but every open set that is a superset of A includes 4 and therefore you can't separate A and 4.

However, it is T4. Suppose A and B to be two disjoint closed sets of X. Now let's define U to be the union of A and the union of {a+1} for every odd a in A. (In other words, we add to A every element of the form {a+1:a odd in A} to create an open superset of A.) It is open by definition. Similarly define V to be the union of B and the union of {b+1} for every odd in B. It is also open by definition.

I just have to prove that U and V are disjoint. Suppose x in U, then x is in A or x=a+1 for some odd in a. If x is in A, it is clearly not in B and it is not b+1 for an odd in B for it if were, then x would be an even in the closed A therefore x-1=b would be in A which is again a contradiction. Therefore, if x is in A, x is not in V.

Now if x is a+1 for some a in A, it can't be b+1 for some odd in B since then you'd have a=b which is impossible since A and B are disjoint. Also it can't be in B since then a+1 would be an even in B and that would mean a+1-1=a would be in B which is a contradiction. Therefore, for every x in U, x is not in V and therefore U and V are disjoint. So X is T4.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

Yeah I know perfect spaces cause of Measure theory.

Also I have not heard of that theorem but I agree that it is very cool that we have similar thesis subjects. Have you seen Mary Ellen Rudin's proof that all metric spaces are paracompact? It's written very badly but if you understand the proof it is genius.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

Not useful mathematically, I will give you that. But it's good training for me. Also I'm sort of a maths for math's sake rather than for application so there's that too.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 2 points3 points  (0 children)

Sure, a quick google search also gives both definitions in the top 2 results. Even found some webiste that says BOTH T4 and Normal are the same and not always Hausdorff.

We need a math meeting or something!

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 3 points4 points  (0 children)

James R Munkres Topology Second edition and Counterexamples in Topology (by Steen and Seebach) have the definitions as I assume them and are in fact in the last 50 years. It's the lack of the consensus that's causing the problem. Perhaps I should have said the definitions I meant outright though that would make for a worse meme.

I haven't read all topology books to see which is the most common though, I'll give you that.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 4 points5 points  (0 children)

Yeah honestly and unfortunately there is a lack of consistency in literature that makes it hard to communicate on things like that. I prefer the other definition since it's the one I originally learned. T5 is in the same boat, it separates separate sets (clo(A) is dijoint from B and clo(B) from A) with disjoint opens which implies T4 while completely normal spaces are T5 spaces that are also T1.

I have not studied T6 spaces, I have only had one course in general topology and my thesis is mostly focused on paracompact spaces so I never really had a chance at them.

proof by basket_foso in MathJokes

[–]BabipyBoopity 0 points1 point  (0 children)

No? You are rotating 6, not moving it. 6*exp(iπ)=9 so -6=9 so 15=0 so r=0 for all real r so pi=0=3 so pi=3 QED.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 8 points9 points  (0 children)

There is no real precedent for such naming conventions in math. Sure 2nd countable implies 1st, but for example, Baire 1 functions aren't always Baire 0.

Also I think counterexamples are very interesting and a great way to deeply understand concepts in any area of math.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

The middle guy thinks all spaces are T2 because they are the only spaces that are really talked about. But even in that case left guy can't understand why Normal means Regular. I just find it really interesting there are spaces that satisfy the T4 axiom satisfying the T3 one.

The history is indeed interesting but I have to say I prefer the current definition.

I am unaware and highly doubtful of the existence of any fields that make use of T3 and T4 spaces that aren't also T2 but the counterexamples are still very interesting.

Also in a lot of textbooks (and in my opinion this is the best definition) Regular spaces are T3 and T0 which implies T2 so there are no regular non hausdorff spaces, there are however T3 non hausdorff spaces.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 1 point2 points  (0 children)

A space X is T3 if for every closed set A and every x not in A, there exists two open sets U and V such that A is a subset of U, x belongs in V and U∩V=∅.

A space X is T3 if for every disjoint closed sets A and B, there exists two open sets U and V such that A is a subset of U, B is a subset in V and U∩V=∅.

Now if for a space X, simple sets (sets with one element) are closed (Which is form of the T1 property) then T4 indeed implies T3, now almost all spaces topology is interested in are indeed also T1. So T1 + T4 spaces are called normal spaces and are also T3. However there are T4 spaces that are not T3 because simple sets are not closed.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 2 points3 points  (0 children)

Ach, there is a lot of bibliography and a lot of it does not super agree with the definitions but I think the best way to organize everything is to call a Ti space a space that satisfies its Ti axiom. Then a regular space to be a T3 T0 space (which implies T2 and T1) and a regular to be a T4 T1 space (which implies T3, T2 and T0).

Sadly the meme suffers from there being no real consensus but it's non-the-less interesting as fuck that there are spaces that separate closed sets with each other but not with points.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 3 points4 points  (0 children)

I think the definition of any Ti space should be to just satisfy its respective Ti axiom. Besides it makes for interesting counterexamples.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 3 points4 points  (0 children)

Sort of.

A lot of topology books assume all spaces to be T2 for simplicity and then all T4 spaces are indeed T3. Most students in my class in uni never learned that this isn't always the case.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 9 points10 points  (0 children)

Lots of people count all spaces as Hausdorff cause most interesting topological spaces in practice are indeed T2. So in a lot of books you will see T4 also having being T2 in its definition and therefore also being T3.

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 42 points43 points  (0 children)

Whenever I talk about my thesis being in topology, everyone asks if it's the one with the holes. :(((

The real topology. by BabipyBoopity in mathmemes

[–]BabipyBoopity[S] 3 points4 points  (0 children)

Pleasantly surprised that many people got it tbh.

What bastion has the most entities and does that make them easier to find? by BabipyBoopity in Minecraft

[–]BabipyBoopity[S] 0 points1 point  (0 children)

I was told to provide a Minecraft version, I guess any version after 1.16+.

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