The question is in the title:

Q1:Is there a topological space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic?

Let us say that $X$ is a *homeomorphic open set space*, or a *hoss* for short, if all the nonempty open subsets of $X$ are homeomorphic.
Such spaces were asked about here, and N. de Rancourt's answer shows that if $D$ is infinite and discrete then $D^\omega$ is a hoss. It follows that every ``ultrametrizable'' space embeds in a hoss. (A space is called *ultrametrizable* if it is homeomorphic to an ultrametric space. Spaces of the form $D^\omega$ are themselves ultrametrizable, and every other ultrametrizable space embeds in one of this form.) This is just about all I know about hosses and spaces that embed in them -- any other information or insight is welcome.

Familiar examples of hosses include the space $\mathbb Q$ of rational numbers and the space $\mathbb R \setminus \mathbb Q$ of irrational numbers.

**EDIT:**

User bof has answered my question by finding a $T_1$ space $X$ containing the real line, and with the property that all nonempty open subsets of $X$ are homeomorphic. However, I still wonder if there is a Hausdorff space with this property:

Q2:Is there a Hausdorff space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic?