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Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...

Let $X$ be a compact metric space and $P$ be a path component of $X$. Since we are not assuming $X$ is locally path connected, $P$ must need not be open nor closed. Certainly, $P$ must be separable ...

Quoting from http://en.wikipedia.org/wiki/End_(topology): "Let X be a topological space, and suppose that K1 ⊂ K2 ⊂ K3 ⊂ · · · is an ascending sequence of compact ...