Can we parameterize a topological space? It has been few months since I started doing topology . There was this idea which struck me a few days ago .
For example the parametrization of a line is $$x=qv+a,$$ where $t$ is the parameter.
Similarly can a topological space be parametrized by another topological space?  
 A: This is called a fiber bundle. It is a map $p:E\to B$ where every point $x\in B$ has an open neighborhood $U$ in $B$ such that $p^{-1}(U)\cong U\times F$. $F$ is called "the fiber" of $p$, since $p^{-1}(x)\cong F$. $E$ is thought of as copies of $F$ parametrized by $B$. There are continuous, smooth, analytic, etc. versions of this definition.
The reason I give this seemingly obtuse definition is that in your example, we aren't just parameterizing one topological space by another; we are parameterizing with $\mathbb{R}$ a space which is homeomorphic to $\mathbb{R}$, meaning both carry the same metric space structure. In particular they carry the same global coordinate $t$, but coordinates are not an intrinsic property of topological spaces. To have global coordinates is to be smoothly embedded in an open subset of Euclidean space and to have local coordinates is to be a manifold. The definition above gives a true characterization of one topological space being parametrized by another, albeit in your case the fiber $F$ is just a point. 
