Dense subsets of Colimit

Question

Let $(X_n,\iota_{n,k})$ be a countable direct system of separable topological spaces, where each $\iota_{n,m}$ is injective. Suppose that $D_n$ is a dense subset of $X_n$; then is $\bigcup_{n} \, \iota_n(D_n)$ dense in $\varinjlim X_n$? Where $\iota_n$ is the canonical (injective) map taking $X_n$ to $\varinjlim X_n$.

Answer 1

The bonding maps $\iota_{n,k} : X_n \to X_k$, $k \ge n$, are determined by the bondings $\iota'_n = \iota_{n,n+1} : X_n \to X_{n+1}$. W.l.o.g. we may assume that $X_n \subset X_{n+1}$ as sets (but not $X_n$ is necessarily a subspace of $X_{n+1}$) and that $\iota'_n$ is the inclusion map. Then as a set $X = \varinjlim X_n = \bigcup_{n=1}^\infty X_n$. The topology on $X$ is obtained as the quotient topology from the disjoint sum $\bigsqcup_{n=1}^\infty X_n$ by identifying $x_n \in X_n$ with $x_i \in X_{n+i}$ for any $i > 0$. Let $p : \bigsqcup_{n=1}^\infty X_n \to X$ be the quotient map.

Let the $D_n$ dense subsets of the $X_n$. We claim that $D = \bigcup_{n=1}^\infty D_n$ is dense in $X$. Let $x \in X$ and $U \subset X$ be an open neigborhood of $x$. Choose $n$ such that $x \in X_n$. Since $p^{-1}(U)$ is open, we see that $U_n = p^{-1}(U) \cap X_n$ is open in $X_n$. Hence $U_n$ contains a point $d \in D_n$. But clearly $d \in D$ and $d \in U$.

Note that we did not use that the $X_n$ are separable. The above result shows, however, that if all $X_n$ are separable, then so is $X$.

Edited:

A more general result is this:

Let $\mathbf X = (X_\alpha, f_{\alpha,\beta},A)$ be any directed system indexed by the directed set $A$ and let $f_\alpha : X_\alpha \to X = \varinjlim \mathbf X$ be the canonical maps. Given dense subsets $D_\alpha \subset X_\alpha$, the set $D = \bigcup_{\alpha \in A} f_\alpha(D_\alpha)$ is dense in $X$.

To see this, note that $X$ is constructed as a quotient of the the disjoint sum $\bigsqcup_{\alpha \in A} X_\alpha$ by identifying $x_\alpha \in X_\alpha$ with $ f_{\alpha,\beta}(x_\alpha) \in X_\beta$ for any $\beta \ge \alpha$. Let $p : \bigsqcup_{\alpha \in A} X_\alpha \to X$ be the quotient map. Then $f_\alpha = p \mid_{X_\alpha}$.

Now let $x \in X$ and $U$ be an open neighborhood of $x$. We have $x = f_\alpha(x_\alpha)$ for some $\alpha$ ands some $x_\alpha \in X_\alpha$. Since $p^{-1}(U)$ is open in $\bigsqcup_{\alpha \in A} X_\alpha$, we have $x_\alpha \in X_\alpha \cap p^{-1}(U)$, the latter being an open subset of $X_\alpha$. Hence there exists $d_\alpha \in D_\alpha$ such that $d_\alpha \in X_\alpha \cap p^{-1}(U)$. This shows that $f_\alpha(d_\alpha) \in D$ and $f_\alpha(d_\alpha) \in U$.