Mathematicians love trying to understand the sense in which things are the same. So, how can we tell that these spaces are really different? I mean, they seem different. Perhaps maybe'm giving away othen ian intuition, but they seem different,. But how do we prove that they're different? Because we've seen that these wildly different spaces are in some sense the same. And so i shall lew, i can give an example of this. The surface of a sphere is a different space than the surface of a doughnut. A taurus and a sphere are different topological spaces. You can have many holed the toris was the doughnut shape. Or you

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