Covering spaces as fiber bundles

Question

I am reading Steenrod. I think there is something wrong or sloppy in his definition of a covering space as a fiber bundle. He writes:

A fibre bundle consists of:

(i) A topological space $B$ (ii) a topological space $X$ (iii) a continuous map $p: B \rightarrow X$ and (iv) a space $Y$ called the fibre. The set $Y_x$, defined by $Y_x = p^{-1}(x)$ is the fibre over $x$. It is required that each $Y_x$ be be homeomorphic to $Y$. Finally, for each $x$ in $X$, there is a neighborhood $V$ of $x$ and a homeomorphism $\phi: V \times Y \rightarrow p^{-1}(V)$ such that $p\phi(x',y)=x'$.

Later he writes:

A covering space $B$ of $X$ is another example of a bundle. The projection is the covring map. The usual definition of a covering space is the definition of bundle modified by requiring that each $Y_x$ is a discrete subspace. and ** $\phi$ is a homeomorphism of $V \times Y_x$ with $p^{-1}(V)$ so that $\phi (x,y)=y$. **. The statement inside ** is obviously nonsense. Is it not?

Answer 1

It might be helpful to think of $Y_x$ as being simply an index set for the decomposition of $p^{-1}(V)$ that occurs in the definition of a covering map.

Let me use the word "sheets" for the elements of that decomposition, i.e. the "sheets" are a decomposition of $p^{-1}(V)$ into open subsets of $p^{-1}(V)$ such that the restriction of $p$ to each sheet is a homeomorphism onto $V$.

So, how do we index the sheets? Answer: the inclusion $Y_x \hookrightarrow p^{-1}(V)$ is a bijection between the set $Y_x$ and the set of sheets: each sheet contains exactly one point of $Y_x$, and each point of $Y_x$ is contained in exactly one sheet. So we can use $Y_x$ itself as an index set. Furthermore, we can topologize this set, giving it the discrete topology (which is no big deal because the subspace topology on $Y_x$ is discrete).

So now define $\phi : X \times Y_x \to p^{-1}(V)$ by letting $\phi(x',y)$ be equal to the unique point in $Y_{x'}$ such that the sheet containing $\phi(x',y)$ equals the sheet containing $y$.

Note: $x$ is a constant in this description, namely a particular point in $V$. I'm using $x'$ as a variable for an arbitrary point in $V$.

Answer 2

Consider a trivial cover : $B = X\times Y$ with $p: B\to X$ the first projection.

Then for all $x\in V$ we can take $V = X$ and we get $\phi_x: V\times Y_x\to p^{-1}(V)$, with $Y_x = \{x\}\times Y$ so $\phi_x: X\times (\{x\}\times Y)\to X\times Y$ defined by $\phi_x(a,(x,y)) = (a,y)$, and $\phi_x(x,(x,y)) = (x,y)$, as expected.