# Let $X$ be a separable Banach space. Let $Y$ be the space formed by adding a point at infinity. Is $Y$ homeomorphic to the unit sphere of $X$?

More specifically, let $Y= X \cup \{\infty\}$ and declare open sets to be the usual open sets in $X$ together with those that are of the form $(X-C)\cup \{\infty\}$, where $C$ is norm closed and norm bounded. So $Y$ is formed in the same way that the one point compactification is when $X$ is finite dimensional.

Clearly if $X = \Bbb R^n$ we have $Y \cong S^n$, which motivates the question. Note that all separable Banach spaces are homeomorphic, so I think $Y$ should not depend on which Banach space is chosen (which makes the Hilbert space the natural candidate to try to work with.)

Any references to a paper/text with a proof or a sketch of a proof (or disproof) are greatly appreciated.

• Your motivating example is incorrect when $n=1.$ – DanielWainfleet Jun 10 '18 at 3:49