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Alekandrov studied with Urysohn, a fellow student, and they produced impressive results. The notion of a compact space and a locally compact space is due to them. They visited Göttingen and impressed Emmy Noether, Courant and Hilbert with their results on when a metric space is metrizable. They also visited Hausdorff in Bonn and Brouwer in Holland in 1926.
Alekandrov became close friends with Hopf and Kolmogorov. He laid the foundations of homotopy theory and was the first to use the phrase 'kernel of a homomorphism'.
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