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Karol Szumiło
Karol Szumiło
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BarratBarratt and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.

Barrat and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.

Barratt and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.

Source Link
Karol Szumiło
Karol Szumiło
  • 7.7k
  • 27
  • 38

Barrat and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.