I have been playing around with the idea of thinking of the set of real valued functions $\mathbb{R}^{\mathbb{R}}$ as the set of uncountably long tuples in $\mathbb{R}$-dimensional space, and I have a question which I have had trouble answering for myself which I would like some help with.
I was...
On $\mathbb{R}$ consider the family $\mathcal{T}$ consisting of $\emptyset,\mathbb{R}$ and al the intervals of type $(-\infty,r)$ with $r\in\mathbb{R}$. This is a topology on $\mathbb{R}$.
Now I have to prove that $\mathbb{R}$, endowed with the euclidean topology is not homeomorphic with $\math...
On $\mathbb{R}$ consider the family $\mathcal{T}$ consisting of $\emptyset,\mathbb{R}$ and al the intervals of type $(-\infty,r)$ with $r\in\mathbb{R}$. This is a topology on $\mathbb{R}$.
Now I have to prove that $\mathbb{R}$, endowed with the euclidean topology is not homeomorphic with $\math...
