Continuous Image of Everywhere Dense Set is Everywhere Dense
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Theorem
Let $A_T = \struct {A, \tau_A}$ and $B_T = \struct {B, \tau_B}$ be topological spaces.
Let $f : A \to B$ be an everywhere continuous surjection.
Let $S \subseteq A$ be everywhere dense in $A_T$.
Then, $f \sqbrk S$ is everywhere dense in $B_T$.
Proof
\(\ds \paren {f \sqbrk S}^-\) | \(\supseteq\) | \(\ds f \sqbrk {S^-}\) | Continuity Defined by Closure | |||||||||||
\(\ds \) | \(=\) | \(\ds f \sqbrk A\) | Definition 1 of Everywhere Dense | |||||||||||
\(\ds \) | \(=\) | \(\ds B\) | Definition 2 of Surjection |
As $\paren {f \sqbrk S}^- \subseteq B$, by set equality:
- $\paren {f \sqbrk S}^- = B$
Therefore, $f \sqbrk S$ is everywhere dense in $B$ by definition.
$\blacksquare$