Here is the definition of a cone (on pg.76) from "Introduction to Homotopy Theory" by Martin Arkowitz:
But the definition of Wikipedia here https://en.wikipedia.org/wiki/Cone_(topology)#:~:text=In%20topology%2C%20especially%20algebraic%20topology,the%20cylinder%20to%20a%20point. is just this $$(A \times I)/(A \times \{0\}).$$
My question is:
What does it mean the union by $\{*\} \times I$ in Arkowitz book? Could anyone explain that for me, please?
Think of the case where $A$ is a circle with a particular point on it (called $\star$) marked red. Extrude it to get $A \times I$, a cylinder that now has a red stripe, $\star \times I$, with the original circle at one end, namely $S^1 \times \{0\}.$ Now collapse that end to a point, and you get a cone, with its tip at the $0$ end and its "base circle" at the $1$ end. And it's got a red stripe as one of the lateral generators.
Arkowitz is proposing to contract that red stripe to a point to get a homotopy-equivalent space, and calling THAT the "cone on $A$". I suppose that doing so makes certain inclusions become nicer maps in some way, so it's just a trick to simplify some later theorem-statements or proofs. But the two spaces are homotopy equivalent, which means that for a homotopy-theory book, they're pretty much the same.
Post-comment addition
The reduced cone is topologically a disk. But suppose that you DOUBLED this process, i.e., made two cones (without shrinking the stripe) and glued them along the circular boundary of each one. You'd get something like a sphere with the two stripes together forming a line of longitude. If you now contract that line of longitude, you get a single "basepoint" for your new sphere, and you've essentially built the "suspension" operation (which will come up a lot) in a way that preserves basepoints really naturally,