Is a fiber in algebra the same as a fiber in topology I am reading Dummit & Foote, and they describe a fiber in algebra as property of a homomorphism. So if $\phi$ is a homomorphism from a group $G$ to a group $H$, then the fibers of $\phi$ are the sets of elements in $G$ that map to a single element in $H$. 
Now I know that there is a notion of fibers in topology as well. I was just wondering if the definition of a fiber in topology is related to the definition of a fiber in algebra? Or is this just a case of inconvenient use of the same name?

Fibre is one of those catch-all words that abounds throughout mathematics. 
The general setup is you have some collection of 'objects' such as groups/rings/topological spaces/manifolds and a class of 'nice' maps between them $-$ group-homomorphisms/ring-homomorphisms/continuous maps/differentiable maps.
In all cases the fibres of the nice map $f \colon X \to Y$ are the subsets $f^{-1}(y)$ for $y \in Y$. We call $f^{-1}(y)$ the fibre over $y$. But this assumes we are talking about a specific map. If we change the map we change what the fibres are. Observe the fibres form a partition of $X$.
You may have heard of something called a fibre bundle. Loosely that means a pair of manifolds $X$ and $Y$ and a map $f \colon X \to Y$ such that all fibres are homeomorphic subsets of $X$.
