Dense

Last updated Nov 1, 2022

Let $(X, \tau)$ be a Topological Space. Then $D \subset X$ is Dense if the Closure of $D$ is $X$.

1. The following are Equivalent
   • $D$ is Dense
   • $X$ is the only closed set containing $D$
   • The Interior of the Complement of $D$ is empty
   • $\forall x \in X$, either $x \in D$ or $x$ is a Limit Point of $D$
   • $\forall x \in X$, every Neighborhood of a Point $U$ of $x$ is not Mutually Disjoint with $D$. That is $U \cap D \neq \emptyset$.
   • $D \cap U \neq \emptyset$ $\forall U \in \tau$
2. Suppose $\mathcal{B} \subset \tau$ is a Topological Basis for $(X, \tau)$. Then The following are Equivalent
   • $D$ is Dense
   • For every $x \in X$, every Basic Neighborhood $B \in \mathcal{B}$ containing $x$ is not Mutually Disjoint with $D$. That is $B \cap D \neq \emptyset$.
   • $D \cap B \neq \emptyset$ $\forall B \in \mathcal{B}$
3. Suppose $X, \tau$ is the Metric Topology generated by Metric Space $(X, d)$. Then The following are Equivalent
   • $D$ is Dense
   • ${x \in X : x = \lim\limits_{n \to \infty} y_{n}, (y_{n}) \subset D} = X$. This is just the Closure of $D$ in a Metric Space.
   • $\forall \epsilon > 0$, $x \in X$, $B_{\epsilon}(x) \cap D \neq \emptyset$. That is, there exists $y \in X$ so that $d(y,x) < \epsilon$.

TODO

• Open Ball