Proposition. Any topological space $X$ can be embedded in a topological space $Y$ with the property that every nonempty open subspace of $Y$ is homeomorphic to $Y.$ Moreover, if $X$ is a
$\text T_1$-space, so is $Y.$

Proof. Let $\kappa$ be an infinite cardinal which is greater than or equal to the number of nonhomeomorphic open subspaces of $X,$ and let $\lambda=\kappa^+.$ (E.g., if $X=\mathbb R,$ take $\kappa=\aleph_0$ and $\lambda=\aleph_1.$)

Let $I$ be an index set with $|I|=\lambda.$ Choose disjoint topological spaces $U_i\ (i\in I)$ so that each $U_i$ is homeomorphic to some nonempty open subspace of $X$ and, for each nonempty open subspace $U$ of $X,$ $|\{i\in I:U_i\text{ is homeomorphic to }U\}|=\lambda.$

Let $Y=\bigcup_{i\in I}U_i$ have the following topology: A nonempty set $W\subseteq Y$ is open just in case $W\cap U_i$ is open in $U_i$ for all $i\in I,$ and $|\{i\in I:U_i\not\subseteq W\}|\le\kappa.$

Proposition. Any topological space $X$ can be embedded in a topological space $Y$ with the property that every nonempty open subspace of $Y$ is homeomorphic to $Y.$ Moreover, if $X$ is a
$\text T_1$-space, it is possible to take $Y$ a $\text T_1$-space as well.

Proof. Let $\kappa$ be an infinite cardinal which is greater than or equal to the number of nonhomeomorphic open subspaces of $X,$ and let $\lambda=\kappa^+.$ (E.g., if $X=\mathbb R,$ take $\kappa=\aleph_0$ and $\lambda=\aleph_1.$)

Let $I$ be an index set with $|I|=\lambda.$ Choose disjoint topological spaces $U_i\ (i\in I)$ so that each $U_i$ is homeomorphic to some nonempty open subspace of $X$ and, for each nonempty open subspace $U$ of $X,$ $|\{i\in I:U_i\text{ is homeomorphic to }U\}|=\lambda.$

Let $Y=\bigcup_{i\in I}U_i$ have the following topology: A nonempty set $W\subseteq Y$ is open just in case $W\cap U_i$ is open in $U_i$ for all $i\in I,$ and $|\{i\in I:U_i\not\subseteq W\}|\le\kappa.$