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Looking at the definitions in https://www.amazon.com/Infinite-Dimensional-Analysis-Hitchhikers-Guide/dp/3540326960/ref=sr_1_1?ie=UTF8&qid=1485902922&sr=8-1&keywords=infinite+dimensional+analysis we can find on page 27 the definition of a neighborhood system to be the collection of all neighborhoods of a point x belonging to some topological space X with a topology t. It suggests that the intersection of any 2 neighborhoods U and V also belongs to the system of neighborhoods, but nothing is mentioned about unions.

My question to you if the following: if U and V are neighborhoods of the same point x, wouldn't their union also be a neighborhood of x and thus belong to the system of neighborhoods?

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    I can't see page 27 in the free sample. But anyway, if $U,V$ are neighborhoods of $x$ then the union $U \cup V$ is also a neighborhood of $x$. Actually, the only thing keeping the collection $\mathcal{O}$ of all neighborhoods of $x$ from being a topology on $X$ is the fact that $\emptyset \notin \mathcal{O}$ – 2017-01-31
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    I see, thank you for your answer. – 2017-01-31

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