What is fiber space?

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 $$B$$.

• It sounds like they're using the phrase "fiber space" to mean fiber bundle.
– anon
Apr 2, 2022 at 3:04
• @anon :But they said that a fiber bundle is a locally trivializable fiber space. Apr 2, 2022 at 3:09
• Probably this then: en.wikipedia.org/wiki/Fibration
– anon
Apr 2, 2022 at 3:17
• Note: every fiber bundle over a paracompact topological space is a Hurewicz fibration, which is in turn a Serre (weak) fibration. Apr 2, 2022 at 11:52
• Do the notes give a definition of a trivializable fiber space? Apr 2, 2022 at 12:14

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.

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.