In this book,as a third completion (the second completion I have not asked it yet as I am still writing it) to The Fundamental group of the circle from "Introduction to knot theory", Ralph H. Fox (1)
The author said:
"The major step in our derivation of the fundamental group of the circle is the following: (5.4) For any continuous mapping $h: E \rightarrow R/3J$ and real number $x \in R$ such that $\phi(x) = h(0,0)$, there exists one and only one continuous function $\bar{h}: E \rightarrow R$ such that $\bar{h}(0,0) = x$ and $h = \phi \bar{h}$. "
And, then the book started to prove it. where:
$E$ is the rectangle consisting of all pairs $(s,t)$ such that $0 \leq s \leq \sigma $ and $0 \leq t \leq \tau,$ where $\sigma$ and $\tau$ are two arbitrary non-negative real numbers.
And $R$ is the real numbers and $J$ is the subring of integers.
Is this the equivalent of the path lifting lemma used in "Allen Hatcher"? Or I am wrong? If so what is the correction?
Thanks!
This is not equivalent to the path lifting lemma, however there is a connection: this lifting statement, and the path lifting lemma, are both special cases of the general lifting lemma.