@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 $\tau_1 \cup \tau_2$
@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.
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.
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.
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.
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.
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).
