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5:22 AM
For instructions how to render MathJax(TeX) in chat see this post on meta.
Perhaps it is worth giving room with this topic one more try.
Although this time I wanted new room to reply to this:
in Mathematics, 16 hours ago, by lyme
Hello. Can someone shorty write why the union of lower limit topology and the upper limit topology isnt a topology?
in Mathematics, 15 hours ago, by Martin Sleziak
The lower limit topology is the one generated by intervals of the form $[a,b)$?
in Mathematics, 15 hours ago, by lyme
yes, $(a,\infty)|a\in R$
in Mathematics, 15 hours ago, by Martin Sleziak
If the union is a topology, then singletons are open.
in Mathematics, 15 hours ago, by Martin Sleziak
So it would have to be discrete.
in Mathematics, 15 hours ago, by lyme
i see thanks :)
in Mathematics, 15 hours ago, by Martin Sleziak
It seems as a sufficient argument. Singletons are open neither in the lower limit nor in the upper limit topology.
in Mathematics, 15 hours ago, by lyme
@MartinSleziak can we say its bacause ie $(-\infty,5)\cup(6,\infty)$ and $(-3,\infty)\cup(-\infty,7)$ are in in $\tau_1 \cup \tau_2$ but intersection of these two subset $(-3,5)\cup (6,7)$ isnt in $\tau_1 \cup \tau_2$
in Mathematics, 14 hours ago, by Martin Sleziak
@lyme What I meant was that $\{1\}=(0,1]\cap[1,2)$ is an intersection of two sets from $\tau_1\cup \tau_2$, but it does not belong to $\tau_1\cup \tau_2$ itself.
in Mathematics, 8 hours ago, by lyme
sorry for interrupting @MartinSleziak thanks. can yuou tell me how can we optain $(0,1]$ and $[1,2) \in \tau_{left}\cup \tau_{right}$
I cant because my book defined these top. as $\tau_r:=\{R,\emptyset\}\cup \{(a,\infty)|a\in \Bbb R\}$ and $\tau_l:=\{R,\emptyset\}\cup \{(-\infty,b)|b\in \Bbb R\}$ so I think $\tau_{left}\cup \tau_{right}$ units should be like $(-\infty,b)\cup (a,\infty)$
So you're asking this @lyme
can yuou tell me how can we optain $(0,1]$ and $[1,2) \in \tau_{left}\cup \tau_{right}$
I cant because my book defined these top. as $\tau_r:=\{R,\emptyset\}\cup \{(a,\infty)|a\in \Bbb R\}$ and $\tau_l:=\{R,\emptyset\}\cup \{(-\infty,b)|b\in \Bbb R\}$ so I think $\tau_{left}\cup \tau_{right}$ units should be like $(-\infty,b)\cup (a,\infty)$
Well, what you wrote as $\tau_r$ is not a topology. Just notice that $\bigcup\limits_{a>0} (a,\infty)=[0,\infty)$, so $\tau_r$ is not closed under unions.
Wikipedia article mentions that for lower limit topology the base is formed by all intervals $[a,b)$.
So maybe you could check what precisely your book says...?
But even if you take the topologies generated by $\tau_r$ and $\tau_l$, then the first one contains sets of the form $[a,\infty)$ and the second one contains sets of the form $(-\infty,b)$, so every half-closed interval of the form $[a,b)$.
So the smallest topology containing $\tau_l$ and $\tau_r$ contains all half-open intervals.
I see that you have posted a related question on main: math.stackexchange.com/questions/839102/…
We can continue the discussion here; or you could post also this on the main if you prefer.
 
6:00 AM
Ignore what I said here:
29 mins ago, by Martin Sleziak
Well, what you wrote as $\tau_r$ is not a topology. Just notice that $\bigcup\limits_{a>0} (a,\infty)=[0,\infty)$, so $\tau_r$ is not closed under unions.
It was a blunder on my side.
So you have shown on the main that $\tau_l$ and $\tau_r$ are topologies.
To see that their union is not a topology, just notice that $(0,1)=(0,\infty)\cap(-\infty,1)$.
The sets $(0,\infty)$ and $(-\infty,1)$ belong to $\tau_l\cup\tau_r$. But their intersection does not. So it is not a topology.
 
 
5 hours later…
11:15 AM
Hi @ThomasKlimpel. Nothing interesting in this room to see as of now I am afraid :-(
If we do not count my embarrassing mistake a few lines above.
 
 
2 hours later…
12:57 PM
I think I understood now how these rooms for general discussions about specific topics work.
The "official" definition of a quasi-topological space seems to be en.wikipedia.org/wiki/Quasitopological_space, but M. Ziegler in "Topological Model Theory" defines a quasitopology on A to be a set of subsets closed under arbitrary unions.
 
You mean this Ziegler's texts: projecteuclid.org/euclid.pl/1235417281
So he calls a set closed under arbitrary unions a quasitopology.
 
Yes, that's the text I'm referring to. He does this, because most of his results generalize to this setting. However, at least the name seem misguided to me, because it already has another definition, and because this sort of space feels quite different from a topological space. Consider products of spaces to see an example. I wonder whether semi-topology would be a better name for such a structure.
 
It will be difficult to find any name that does not have an assigned meaning. Google: "semi-topological space".
Product would be the quasi-topology generated by "open rectangles" $A\times B$?
The name quasi-topology only appears two or three times in that text?
So he's just saying something like: This is the setting where everything from previous sections works. But no details are given there, right?
In any case, at the fist glance something like this seems like a rather strange structure.
In fact answer to this question gives a different description of quasi-topology: Is an “open system” just a topological space?. (At least if the proof in the answer is correct.)
 
1:24 PM
This product would give a "nice" monoidal structure, but the product in the sense of category theory gives something much less nice. Basically the topology on the product is "too coarse" to be of much use (at least that's my impression).
 
So the product would be generated by set of the type $A\times Y$ and $X\times B$.
(The product in the categorical sense.)
 
The name "open system" sounds good to me, at least better than quasitopological space.
 
 
2 hours later…
3:10 PM
You wrote: I think I understood now how these rooms for general discussions about specific topics work.
To be honest, I am not sure how exactly are they supposed to work. But I still tihnk they might be useful.
 
3:23 PM
room topic changed to General topology: For any discussions about general topology [general-topology]
 

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