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In Spanier's, Algebraic Topology, he writes:

"A topological pair $(X,A)$ consists of a topological space $X$ and a subspace $A \subset X$."

In a question at the end of the section he asks a question about the topological pair $(CX,X)$, where $CX$ is the cone over $X$. Why is $X$ a subspace of $CX$?

user1058860
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1 Answers1

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The cone over $X$ is typically defined as the quotient space $CX=X\times[0,1]/X\times\{1\}$. The subspace $X\times\{0\}\subseteq CX$ is homeomorphic to $X$, and "$(CX,X)$" is just a small abuse of notation for $(CX,X\times\{0\})$.

Eric Wofsey
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  • Alright I have a small follow up question. So Spanier also has a problem where he states about showing $X$ is a retract of $CX$ for some conditions and he also writes $i : X \subset Z_{f}$ where, $Z_{f}$ is the mapping cylinder. In these cases $X$ is also meant as $X \times 0$, right? – user1058860 Dec 30 '15 at 04:49
  • Yes, something like that (I can't say for sure without knowing what $f$ is and exactly how he defines $Z_f$). – Eric Wofsey Dec 30 '15 at 04:56
  • He defines the mapping cylinder in the following manner: "Let $f: X \rightarrow Y$ and let $Z_{f}$ denote the quotient space obtained from the topological sum of $X \times I$ and $Y$ by identifying $(x,1) \in X \times I$ with $f(x) \in Y$. $Z_{f}$ is called the mapping cylinder of $f$..." – user1058860 Dec 30 '15 at 04:57
  • When I mentioned $i$ Spanier defined it as the imbedding $i: X\rightarrow Z_{f}$, for some $f:X\rightarrow Y$, with $i(x) = [x,0]$ ($i$ is later denoted as an inclusion map $i:X \subset Z_{f}$). – user1058860 Dec 30 '15 at 05:01
  • Right, with that definition the standard way to consider $X$ as a subspace of $Z_f$ is as $X\times{0}$. – Eric Wofsey Dec 30 '15 at 05:07