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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 ?

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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.

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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.

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  • $\begingroup$ Hi professor Brandsma, could I ask your assistance here? I think I found a good answer but I'm not sure about. $\endgroup$ Jun 14, 2020 at 21:15
  • $\begingroup$ @AntonioMariaDiMauro I'm honoured but I'm not a professor. Teacher, yes. It's a protected title (at least in the Netherlands). $\endgroup$ Jun 14, 2020 at 22:23
  • $\begingroup$ 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. $\endgroup$ Jun 15, 2020 at 8:35
  • $\begingroup$ @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. $\endgroup$ Jun 15, 2020 at 8:39
  • $\begingroup$ I'm studing engineering and to study Fluid Dynamics I have to learn Differential Geometry and so even Topology and Set Theory. $\endgroup$ Jun 15, 2020 at 8:46

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