# classification of covering spaces

Let $X$ be a connected^{}, locally path connected and semilocally simply
connected space. Assume furthermore that $X$ has a basepoint $*$.

A covering $p:E\to X$ is called *based* if $E$ is endowed with a
basepoint $e$ and $p(e)=*$. Two based coverings ${p}_{i}:{E}_{i}\to X$, $i=1,2$ are called
equivalent^{} if there is a basepoint preserving equivalence $T:{E}_{1}\to {E}_{2}$ that
covers the identity^{}, i.e. $T$ is a homeomorphism and the following diagram
commutes