Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
The Theorem is the following.
" X be a space. Then $X$ is locally compact Hausdorff iff there exists a space $Y$ satisfying the following conditions
1) $X$ is a subspace of $Y$.
2) The set $Y-X$ consists of single point.
3) $Y$ is a compact Hausdorff space. "
$X$ is a subset of $Y$ . Then why $X$ is not becoming naturally a subspace of $Y$?
Can anyone please make me understand ?
You start with a space $\mathbf X=(X,\tau_X)$ where $X$ denotes the underlying set and $\tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $\mathbf Y=(Y,\tau_Y)$ might exist such that $X\subseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,\rho)$ where $\rho$ denotes the subspace topology on $X$ inherited from $\mathbf Y$.
So it says nothing yet about our original space $\mathbf X=(X,\tau_X)$.
To make that change it must be demanded that $\tau_X$ and $\rho$ coincide or equivalently that $\mathbf X=(X,\tau_X)$ is a subspace of $\mathbf Y$.