What is fiber space?

Question

I'm reading a notes about algebraic topology.It uses a phrase 'fiber space' without any definition.I think maybe it's a generalization of fiber bundle.So I want to ask if there's anyone who knows what it means.

The following are some statements on the notes about 'fiber space'.

1. A fiber bundle over a topological space $B$ is a locally trivial fiber space,say, a $B$-space $(E,p)$ satisfying:for any $b \in B$, there is a neighborhood $V$ of $b, s.t. (p^{-1}(V),p|_{p^{-1}(V)})$ is trivializable.(A $B$-space $(E,p)$ here means a pair $(E,p)$ where $E$ is a topological space and $p:E\longrightarrow B$ is a continuous map.)
2. Fiber space over any close interval [a,b] is trivializable.
3. If $(E,p)$ is a fiber space over topological space $B$,then $\{\ b\in B: p^{-1}(b)=\varnothing\}\ \ and \ \{\ b\in B: p^{-1}(b)\neq\varnothing\}=p(E)$ are both open in $B$.

Thanks in advance!

Answer 1

It is most likely a fibration. Have a look at this. It is certainly not necessarily a fiber bundle (which is defined as a locally trivial fiber space). Here are three references.

1. WolframMathworld: A fiber space, depending on context, means either a fiber bundle or a fibration.

2. Encyclopedia of Mathematics: An object $(X,π,B)$, where $π:X→B$ is a continuous surjective mapping of a topological space X onto a topological space B (i.e., a fibration). Note that $X, B$ and $π$ are also called the total space, the base space and the projection of the fibre space, respectively.
   Note that this definition is flawed because a fibration is not the same as a continuous surjective mapping.

3. Hurewicz, Witold. "On the concept of fiber space." Proceedings of the National Academy of Sciences of the United States of America 41.11 (1955): 956-961.
   Hurewicz has a very special definition of "fiber space", but proves that his fiber spaces are fibrations in the modern sense.

Also see What is fiber space？Is it related to the fiber bundle？

Concerning your property 2 see Pavešić, Petar. "A note on trivial fibrations." Glasnik matematički 46.2 (2011): 513-519.

However, it seems that your property 3 is not satisfied. Take for example the closed topologist's sine curve $S$. The inclusion of the oscillating part $O = \{(x,\sin(1/x)) \mid x \in (0,1] \}$ into the closed topologist's sine curve is a fibration, but the set of points of $S$ having a non-empty fiber is not open. See also Surjectivity of Hurewicz fibrations.