VIP topology on $X$

general-topology

I came across the following: Consider the VIP topology on $X$:

$$\tau=\{U \subset X: U=\emptyset \;\;\text{or}\;\;a \in U \}$$ where $a \in X$ is fixed.

The author proves every non empty open set is dense in $X$. After he proves every proper subset is nowhere dense.

How the second one is true ? because take any proper non empty open set $U$, and using first one, $U$ is dense. But by the second one $U$ is nowhere dense. What's going on? Is I'm misunderstand something or it is a typo in a book?

I add a reference to it: See 4 and 7

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Here's his proof:

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Best Answer

The proof you post clearly shows that we assume that $\overline A$ is a proper subset of $X$, rather than just $A$ itself.

So yes, this is in fact a typo. In other words, the proof shows that if $A$ is not dense, then it is nowhere dense. Which is to be expected.

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