Difference between dense set and everywhere dense set
I'd like to know what is the main difference between these definitions? In functional analysis we can face everywhere dense set but there is no the same definition in topology. In Engelking "General topology" we have only dense, dense-in-itself and nowhere dense set. I'm just a little bit confused about definitions.
For a set $A \subseteq X$, a space:
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dense = everywhere dense $\iff$ the closure of $A$ is $X$ $\iff$ every non-empty open subset of $X$ intersects $A$.
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dense-in-itself means that $A$ has no isolated points in its subspace topology $\iff$ There is no open subset $O$ of $X$ so that $|O \cap A|=1$ $\iff$ $A \subseteq A'$ (where the latter is the set of limit points of $A$, the so-called derived set).
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nowhere dense means there is no non-empty open set $O$ so that $O \subseteq \overline{A}$ $\iff$ there is no non-empty open set $O$ so that $O \cap A$ is dense in $O$ (this explains the name; in classic parlance "nowhere" meant "inside no non-empty open set"). There are several other equivalences I won't bore you with.
But usually everywhere dense is an old-fashioned synonym for what is just "dense" so the definition under 1.
Kolmogorov's definition of $A$ is dense in $B$ is just saying that $A \cap B$ is dense in $B$ in its subspace topology, by standard facts on the closure. Or more intuitively: we can approximate points of $B$ arbitrarily closely by points of $A$. "everywhere dense" is just emphasising that we can do this for all points of $X$ now.