Problem of the Week #118 - September 1st, 2014

[Chris L T521]

Here's this week's problem!

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Problem: Let $H$ denote a group that is also a topological space satisfying the $T_1$ axiom. Show that $H$ is a topological group if and only if the map of $H\times H$ into $H$ sending $x\times y$ into $x\cdot y^{-1}$ is continuous.

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[Chris L T521]

This week's problem was correctly answered by Euge and mathbalarka. You can find Euge's solution below.

[sp]Let $\mu, \psi : H \times H \to H$ be given by $\mu(x, y) = xy$ and $\psi(x, y) = xy^{-1}$. Define $\phi : H \to H$ by the equation $\phi(x) = x^{-1}$. Suppose $H$ is a topological group. Then $\mu$ and $\phi$ are continuous, whence the composition $\psi = \mu \circ (1 \times \phi)$ is continuous. Conversely, suppose $\psi$ is continuous. Since $H$ is T1, the set $\{1\} \times H$ is a closed subspace of $H \times H$. Hence $\phi$, being the restriction of $\psi$ to $\{1\} \times H$, is continuous. So $\mu$, being the composition of continuous maps $\psi$ and $1 \times \phi$, is continuous.[/sp]