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For any topological space $X$, as the title explains, how many ways to construct a dense subspace of $X$? For example, we can construct a dense subspace which is the union of disjoint open subsets of $X$.

Added: If I may ask more, if $X$ is compact, do we have more ways to construct a dense subspace of $X$?

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Your edit doesn't help at all. I still don't know what you mean by "how many". It looks like you want examples that are "natural", explicit constructions of different flavor. But without an explanation of what you really mean by "how many (different?) ways", there is no way to answer this question. –  mrf Jan 21 '13 at 10:03
    
Are you simply asking for methods of constructing dense subsets? –  Arthur Fischer Jan 21 '13 at 11:11
    
@ArthurFischer I hope the dense subsets we constructed have topological properties. Just as the question talked, for example, a dense subspace which can be the union of disjoint open subsets of $X$. –  Paul Jan 21 '13 at 11:18

2 Answers 2

Hint: Every compact metric space has countable base and is seperable, i.e it has a countable dense subset.

for $n\in \mathbb{N}$ the open sets $N_{1/n}(x)$ for $x\in X$ forms a open cover of $X$. since $X$ is compact, chhose a finite subcover $\{N_{1/n}(x_{n,1}),\dots,N_{1/n}(x_{n,k_n})\}$, note that for each $n$ the collection of $x$'s is different; therefore they must be labelled by both $n$ and a second parameter.

consider the countable base for $X$ $$\mathfrak{B}=\{N_{1/n}(x_{n,j}:n\in\mathbb{N}, 1\le j\le k_n\}$$ , now take any open set $U\subseteq X$ chose $\epsilon>0$ such that $N_{\epsilon}(x)\subseteq U$....enough?

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(Assuming you mean "how many" in the sense of cardinality.)

If $X = \mathbb{R}$ (or a subset of $\mathbb{R}$), there are $2^c$ dense subsets, where $c$ is the cardinality of the continuum.

Let $A$ be a countable dense subset of $X$ (for example, an enumeration of the rational points in $X$). Then $A \cup B$ is dense for any $B \subset X \setminus A$, so by varying $B$ you get $2^c$ different possibilities. On the other hand, there are "only" $2^c$ subsets of $X$, so the number of dense subsets can't be larger than this.

(You should be able to generalize this to many other toplogical spaces.)

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Thanks for your answer. However, you may misunderstand my question. I don't want to discuss the cardinality. Thanks all the same. –  Paul Jan 21 '13 at 9:48
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@Paul what exactly do you mean by "how many ways to construct a dense subspace" then? –  mrf Jan 21 '13 at 9:50
    
See the question re-edited. –  Paul Jan 21 '13 at 9:59

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