Real line as a product topological space Let $\Bbb R$ be endowed with the standard topology. Clearly, we can trivially represent
$$
  \Bbb R \cong \Bbb R^0\times \Bbb R \tag{1}
$$
and also, there is not such topological space $X$ that $\Bbb R \cong X^2$. Thus, I wonder whether necessary
$$
  \Bbb R \cong X\times Y
$$
implies that $X = \Bbb R^0$ and $Y = \Bbb R$ (or vice-versa). With $\Bbb R^0$ I mean the singleton topological space $1$.

t.b.'s answer to your previous question would seem to work mutatis mutandis in this case as well:

If $\mathbb{R} = X \times Y$, then both $X$ and $Y$ are path connected.  If both $X$ and $Y$ have cardinality $> 1$ it follows that removing any single point from $X \times Y$ yields a path connected space, but the real line clearly does not have this property.

