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Mike Skirvin
Mike Skirvin
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You can use connectedness to prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^n$ for any n>1 by noting that $\mathbb{R} \backslash 0$ is not connected while $\mathbb{R}^n \backslash 0$ is not connected.

The students may not be very impressed by this as it is telling them something they probably already assumed was true. I suppose that if you wanted them to discover cohomology, you could challenge them to find a reason why Euclidean spaces of different dimensions are never homeomorphic (I realize that this probably isn't very reasonable).

You can use connectedness to prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^n$ for any n>1 by noting that $\mathbb{R} \backslash 0$ is connected while $\mathbb{R}^n \backslash 0$ is not connected.

The students may not be very impressed by this as it is telling them something they probably already assumed was true. I suppose that if you wanted them to discover cohomology, you could challenge them to find a reason why Euclidean spaces of different dimensions are never homeomorphic (I realize that this probably isn't very reasonable).

You can use connectedness to prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^n$ for any n>1 by noting that $\mathbb{R} \backslash 0$ is not connected while $\mathbb{R}^n \backslash 0$ is connected.

The students may not be very impressed by this as it is telling them something they probably already assumed was true. I suppose that if you wanted them to discover cohomology, you could challenge them to find a reason why Euclidean spaces of different dimensions are never homeomorphic (I realize that this probably isn't very reasonable).

Source Link
Mike Skirvin
Mike Skirvin
  • 2.7k
  • 1
  • 19
  • 18

You can use connectedness to prove that $\mathbb{R}$ is not homeomorphic to $\mathbb{R}^n$ for any n>1 by noting that $\mathbb{R} \backslash 0$ is connected while $\mathbb{R}^n \backslash 0$ is not connected.

The students may not be very impressed by this as it is telling them something they probably already assumed was true. I suppose that if you wanted them to discover cohomology, you could challenge them to find a reason why Euclidean spaces of different dimensions are never homeomorphic (I realize that this probably isn't very reasonable).