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Dan Petersen
Dan Petersen
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Of course not. Let $X = H \times S^1$ where $H$ is contractible and infinite dimensional$X = \mathbf R^\infty \times S^1$, and let $X_n = H \times I_n$$X_n = \mathbf R^{n-1} \times I_n$ where $\{I_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.

Of course not. Let $X = H \times S^1$ where $H$ is contractible and infinite dimensional, and let $X_n = H \times I_n$ where $\{I_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.

Of course not. Let $X = \mathbf R^\infty \times S^1$, and let $X_n = \mathbf R^{n-1} \times I_n$ where $\{I_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.

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Dan Petersen
Dan Petersen
  • 40.2k
  • 2
  • 114
  • 201

Of course not. Let $X = H \times S^1$ where $H$ is contractible and infinite dimensional, and let $X_n = H \times I_n$ where $\{I_n\}$ is an increasing sequence of intervals in $S^1$ whose union is $S^1 \setminus \{\mathrm{point}\}$.