I am going through an introductory topology chapter and finding some difficulty in the following question. I had a chance to go through Daniel's and Brian's answers on this page. However, I am still a bit not convinced.
If $D_1$ is an open dense set in $\mathbb R^p$ and $D_2$ is any dense set in $\mathbb R^p$, then,Show that $D_1 \cap D_2$ are also dense in $\mathbb R^p$ .
Attempt: Given that $D_1$ is an open dense set in $\mathbb R^p$ which means if $x \in D_1^c$, then every neighborhood of $x$ contains at least one element of $D_1$.
But, this fact is contradictory because elements of $D_1$ are not supposed to be in every neighborhood of elements in $D_1^c$.
$(a)~$Hence, can we infer that $D_1^c$ must not have any non void open set else any open ball in $D_1^c$ will also contain elements of $D_1$ which is a contradiction?
Suppose that $D_1 \cap D_2$ is not dense in $\mathbb R^p$. Then, for some $y \in \mathbb R^p$, there is a neighborhood of $y = B(y,r)$ which does not contain any element of $D_1 \cap D_2$.
This is certainly possible because $B(y,r)$ can contain elements of $D_1/ (D_1 \cap D_2)$ and $D_2/ (D_1 \cap D_2)$.
How do I move forward?
Thank you for your help