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I am revising for algebraic topology and an exercise sheet from class is just weird straight. Here's the problem I don't even know what it means

Let $X$ be a space. Let $Y$ be a space obtained from $X$ by adding a whisker at $ a \in X$. More formally, $Y$ is the quotient space of the disjoint union $X \cup I$ in which $a \in X$ is identified with $0 \in I$. Prove that inclusion $i:X \to Y$ is a homotopy equivalence.

What is a whisker. after $20$ years of math I've never heard of that term apart fro "box and whiskers" and it just doesn't make sense to me here. And the "more formally," bit, "$a$ is identified with"? What does it mean "identified with." I think the closest possible meaning would be "defined by" but the $0 \in I$ that follows doesn't make sense if I replace "identified with" by "defined by".

algebraic topology is obscure enough in its own regard and I'm just hoping professors would stop making the problem statement obscure as well. Is anyone familiar with these strange terminologies?Can you explain them to me?

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  • $\begingroup$ I think he means whisker as in a piece of hair. He's gluing an interval $I$ to $X$ by gluing the zero end of it to the element $a$. The other end is free, so the interval literally becomes a whisker. $\endgroup$
    – MonadBoy
    Mar 29, 2016 at 15:54
  • $\begingroup$ Are you familiar with the quotient topology? $\endgroup$
    – Alex Provost
    Mar 29, 2016 at 16:03
  • $\begingroup$ It's not a formal term. It just looks like a whisker. A line segment sticking out. $\endgroup$
    – Akiva Weinberger
    Apr 10, 2016 at 4:35

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The problem states that we have the space $X\cup I/\sim$, where the equivalence relation is qiven by $a\sim 0$, where $a\in X$ and $0\in I$. What this graphically means is that we add a shape in the form of a hair (or a whisker) to the space $X$. We do this by sticking one end of the hair to the space $X$. Note that you get a homotopy equivalence by "shrinking" the length of the whisker.

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  • $\begingroup$ Is the hair or whisker always contractible? When does a whisker imply a homeomorphism? $\endgroup$
    – Sam Hoffman
    May 21, 2022 at 8:44

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