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Can you please see the following question : Let X be any set with three elements or more and B be the collection of all two element subset of X . Show that B is not a base for any topology >>

I think on it as the following : since there are more than three elements in X then set B must have two subset of X of the form {x,y} , {y,z} suppose by contrary B is base then {x,y} , {y,z} are open subset their intersection {y} is open but it can not generated from B so contradiction ..

is it true ?

José Carlos Santos
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Aya
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Yes, this is correct. You showed that a topology generated by $B$ must be discrete. But singeltons are not a union of elements in $B$, so this is impossible.

J. De Ro
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Let $\{x,y,z\}$ be three distinct elements from $X$.

So $B_1 = \{x,y\}$ and $B_2 = \{y,z\}$ are in the base $\mathcal{B}$. In particular that are open so $B_1 \cap B_2 = \{y\}$ is open.

But there cannot be any $B \in \mathcal{B}$ such that $y \in B \subseteq B_1 \cap B_2$ as $2 = |B| > |B_1 \cap B_2|=1$.

So $\mathcal{B}$ is not a base for a topology.

Henno Brandsma
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  • Hi professor Brandsma, could I ask your assistance here? I think I found a good answer but I'm not sure about. – Antonio Maria Di Mauro Jun 14 '20 at 21:15
  • @AntonioMariaDiMauro I'm honoured but I'm not a professor. Teacher, yes. It's a protected title (at least in the Netherlands). – Henno Brandsma Jun 14 '20 at 22:23
  • Excuse me for the audacity. I'm study naval engineering at the university Federico II in Naples: unfortunately in Italy the teachers are terribly lazy and andditionaly they teach the Math as the lived in the XIX century neglecting totally Set Theory and Topology to explain any analytic topic so that I'm studying alone as autodidact and so from this you can understand that for me you are a professor because in these months you have explained to me a lot of things. Anyway if you prefer I can only call you theacher or professor. – Antonio Maria Di Mauro Jun 15 '20 at 08:35
  • @AntonioMariaDiMauro I’m fine with that. What does a naval engineering student want to do with set-theoretic topology? Italy was in the 19th century one of the places where topology was “invented “ (or discovered) so it’s fitting they still teach it that way, maybe. – Henno Brandsma Jun 15 '20 at 08:39
  • I'm studing engineering and to study Fluid Dynamics I have to learn Differential Geometry and so even Topology and Set Theory. – Antonio Maria Di Mauro Jun 15 '20 at 08:46
  • @AntonioMariaDiMauro you don’t really need topology for that though (not the kind you study anyway) but real analysis etc. In topology differentiability is not studied. Just very global properties. – Henno Brandsma Jun 15 '20 at 09:22
  • The fact is that with the assistance of Topology the dimostration of differentiation and integration theorem are more rigorous and more simple: for example using induction and the universal mapping theorems for products it is possible to prove that a vectorial field is of class $C^r$ if and only if its components are of class $C^r$. – Antonio Maria Di Mauro Jun 15 '20 at 09:27
  • @AntonioMariaDiMauro that fact is also easy to see using analysis arguments. – Henno Brandsma Jun 15 '20 at 09:28
  • Umm .... perhaps I have to explain my story. For a long time I hated Math: at school I always had low grades in Math so that initially I decided to dedicate myself to humanistic studies but then for different reasons I decided to dedicate myself to scientific studies so I started to study naval engineering. – Antonio Maria Di Mauro Jun 15 '20 at 10:07
  • However when I attended the first math lessons at univerity with wonder I discovered that I couldn't learn much with the training method of my professors. – Antonio Maria Di Mauro Jun 15 '20 at 10:31
  • Initially I thought that the Math was excessively difficult to me but then I understood that the Math is not a natural science but a human construction created to describe some natural phenomenons so I started to research a more simple training method to learn it (any phenomenon could be described through many different way!!!) and so I discovered Set Theory and Topology through wich I finally understood many thinghs that my professors thought difficul to understand for a boy. – Antonio Maria Di Mauro Jun 15 '20 at 10:31
  • Therefore I understood that the Math is a human creation and so any human can understand it: the problem is to find the training method more appropriated for yourself!!! That's all. – Antonio Maria Di Mauro Jun 15 '20 at 10:31