Let $ \Omega$ be the set of ordinales with $\omega_1$ being the first uncountable ordinal. Denote $\Omega ( \omega) = \mathbb{N} \cup \lbrace \omega \rbrace$, where $\omega$ is the first infinite ordinal. When $\Omega(\omega)$ is given its order topology, the points of $\mathbb{N}$ are isolated and the point $\omega$ has for basic nhoods the sets $\lbrace n,n+1,...\rbrace \cup \lbrace \omega \rbrace $.
The product space $\Omega \times \Omega(\omega)$ will be denoted T$^{*}$. Considere the subspace $T=T^{*}-\lbrace (\omega_1,\omega) \rbrace$ of T$^{*}$.
Show that the one-point compactification of $T$ is $T^{*}$.
I know that I must prove that the topology on $T^{*}$ coincides with the topology obtained by the compactification.
I think it is enough to study the basic neighborhoods of the elements of $\Omega \times \Omega(\omega)$.
However I've some problems understanding how I must do that.
Can anyone help me?
Theorem: if $X$ is compact Hausdorff and $p \in X$ is a non-isolated point of $X$, then $X$ is the one-point compactification of $X\setminus\{p\}$ (or $X - \{p\}$ if you prefer that). This is the essential uniqueness of the one-point compactification.