I am reading the book "Algebraic topology" by Tammo Tom Dieck and I am having a problem with a particular notation. I have attached the screenshot from that particular page. In the last line what is meant by $\text{TOP}_B$. The category of topological space is denoted as TOP, but I don't know what is $\text{TOP}_B$. Can anyone explain?
That is an example for a slice category. The objects are all arrows in the category of topological spaces with codomain $B$ and for two arrows $f \colon X \rightarrow B$ and $g \colon Y \rightarrow B$ the class of morphisms $\text{Hom}(f,g)$ in the slice category is given by all arrows $h \colon X \rightarrow Y$ in the category of topological spaces such that $f = g \circ h$.
For example the category of pointed topological spaces can be seen as the category of topological spaces under a point (if you fix a domain). Analogously for the category of $R$-algebras over a given ring $R$.
This means you have probably already seen coslice categories and the category you are asking for is given by a dual notion.
In tom Diecks's book you find a detailed explanation in chapter 2.2 "Further Homotopy Notions", page 32.