Why we need a topological extension of a topological space? We extend topological spaces. One point compactification is an example. But why we need this type of extension?
 A: Philosophically, we study an extension $\widehat{X}$ of a space $X$ because $\widehat{X}$ tells us something about the space $X$, but $\widehat{X}$ is "nicer" than $X$ in some way. "Nicer" sometimes means that invariants are easier to compute for $\widehat{X}$ than they are for $X$.
Here is perhaps the most standard application of the one point compactification. 

Theorem. Let $n$ and $m$ be positive integers. Then $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^m$ if and only if $n=m$.

Here is a sketch of the proof.


*

*If $X$ and $Y$ are homeomorphic topological spaces, then their one point compactifications $\widehat{X}$ and $\widehat{Y}$ are also homeomorphic.

*The one point compactification of $\mathbb{R}^n$ is homeomorphic to the $n$-sphere $S^n$ defined as
$$S^n = \{ \mathbf{x}\in\mathbb{R}^{n+1}~:~ |\mathbf{x}|=1\}.$$

*The (simplicial) homology of $S^n$ is
$$H_i(S_n) \cong \begin{cases}
\mathbb{Z} & i=0~\text{or}~n,\\
0 & \text{else.}
\end{cases}$$

*Homology is a homeomorphism invariant. Thus $S^n$ and $S^m$ are not homeomorphic if $n\neq m$, which implies the theorem.

