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Consider a continuous map $f: X \rightarrow Y$, where $X$ and $Y$ are compact $CW$ complexes. Are there interesting examples of $f$ such that the pre-image of a cell in $X$ is not a cell in $Y$?

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What's an interesting example? – Qiaochu Yuan Feb 16 '14 at 8:44
Any map can satisfy this if you are willing the change the CW structure of $X$. Just take the cells and start subdividing, then the preimages won't be cells anymore. For example, take $X = S^1$ with two $0$-cells and two $1$-cells, then take $Y = S^1$ with one $0$-cell and one $1$-cell, and consider the identity map $S^1 \to S^1$. – Justin Young Mar 13 '14 at 14:01

Why not? Take any continuous map $f:X\to Y$ and if it doesn't already have this property, perturb it slightly. For instance, if we realize $\mathbb{S}^1$ with one zero-cell at $1$ and one one-cell connecting $1$ to itself, any continuous self-map which does not take $1$ to itself meets your criteria.

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