Let $X$ and $Y$ be two CW complexes. Is it true that any homeomorphism $f:X\longrightarrow Y$ is a cellular map, meaning that $f(X^n)\subset Y^n$ ? where $X^n$ refers to the $n-$skeleton of $X$.
2026-03-25 14:21:28.1774448488
A homeomorphism between two CW complexes is a cellular map
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No. The interval $[0,1]$ can be given CW-structures $X$ such such that $X^0 =\{ \{0\},\{1\}\}$ and $Y$ such that $Y^0 = \{\{0\},\{1/2\},\{1\}\}$. Then $id : Y \to X$ is not cellular.