lifting criterion for continuous maps

Overview

Given a cover and a continuous map from a topological space to the base, a natural question to ask would be if the map can be lifted to the cover.

Statement

Let $f:(Y,y_0)\to (X,x_0)$ be a continuous map of pointed spaces and $p:(E,e_0)\to (X,x_0)$ a cover.

There exists a lift of $f$ to $E$ if and only if $f_{\ast }{\pi }_1(Y,y_0)$ is a subgroup of $p_{\ast }{\pi }_1(X,x_0)$ (the induced subgroup from cover).

Consider the following diagram:

Proof

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