Difference in the definitions of density in topology?

Question

I have been trying to understand the concept of density and have so far come across two definitions. That is, the set $A$ is dense in a space $B$ if

cl(A) = B,

however, I have also come across the following definition.

The set $A$ is dense in $B$ if

$A \subset B$ and $B \subseteq cl(A)$.

Just wondering if the two definitions are indeed the same.

Also, additionally, we have it given that

$B \subseteq cl(A)$.

Taking the closure of both sides yields

$cl(B) \subseteq cl(A)$.

I am also just wondering if this is also indeed true.

Thanks for the help.

Answer 1

So these are slightly different notions linguistically:

the first is the more standard one and is for spaces $X$: $A$ (a subset of $X$) is called dense in $X$ when $\operatorname{cl}(A) = X$.

The second notion is for subsets of a space $X$ : the set $A$ is called dense in $B$ whenever $A \subseteq B$ and $B \subseteq \operatorname{cl}(B)$.

But this second notion is not really different in content: it means that $\operatorname{cl}_B(A) = A$, where $\operatorname{cl}_B$ is the closure operator in $B$ with the subspace topology seen as a space in its own right.

This can be seen by applying the following general fact for subspaces and their closure operator, and how it relates to the closure in the ambient space:

Fact: If $B$ is a subset of $X$ in the subspace topology w.r.t. $X$, and $A \subseteq B$ then $\operatorname{cl}_B(A) = \operatorname{cl}(A) \cap B$; or in words: the closure of $A$ in the subspace $B$ is just the part of the full closure of $A$ (in $X$) that lies inside $B$.

This fact can be proved directly from the definition of the subspace topology and the characterisation of the closure of $A$ as all points $x$ such that all neighbourhoods of $x$ intersect $A$, or from the fact that closed sets of $B$ are closed sets of $X$ intersected with $B$, and the closure of $A$ is the smallest closed set that contains $A$.

Now suppose $A \subseteq B$ and $A$ is dense (in the first sense) in the subspace $B$, so $\operatorname{cl}_B(A) = B$ and by the fact we can rewrite this as

$$\operatorname{cl}_B(A) = \operatorname{cl}(A) \cap B = B$$ but simple set theory tells us that $$C \cap B = B \text{ iff } B \subseteq C$$ for all $B,C$ and so we actually get that

$$\operatorname{cl}_B(A) = B \text{ iff } B \subseteq \operatorname{cl}(A)$$

So the "notions" of dense mean the same thing: $A$ is dense in $B$ if we can "approximate all points of $B$ by points of $A$", or $B$ is a subset of $\operatorname{cl}(A)$ (where $\operatorname{cl}(A)$ can be thought of as "all points that can be approximated by points of $A$", intuitively (or more formally e.g. if you use filters or nets)