Local topological group

A local topological group or local group is a topological space which behaves like a topological group sufficiently close to a distinguished identity element. topology Formally a local group $(𝑋, T, 𝑒, Θ, Ω, 𝑖, 𝑚)$ consists of a topological space $(𝑋, T)$ , open subets $Θ \subseteq 𝐺$ and $Ω \subseteq 𝐺 \times 𝐺$ , a distinguished identity element $𝑒 \in Θ \subseteq 𝐺$ , and continuous functions $𝑖 : Θ \to Θ$ and $𝑚 : Ω \to 𝐺$ such that

$(𝑒, 𝑔), (𝑔, 𝑒) \in Ω$ and $𝑚 (𝑒, 𝑔) = 𝑚 (𝑔, 𝑒) = 𝑔$ for all $𝑔 \in 𝐺$ (identity)
if $(𝑔, ℎ), (ℎ, 𝑡), (𝑚 (𝑔, ℎ), 𝑡), (𝑔, 𝑚 (ℎ, 𝑡)) \in Ω$ then $𝑚 (𝑚 (𝑔, ℎ), 𝑡) = 𝑚 (𝑔, (ℎ, 𝑡))$ (associativity)
$(𝑔, 𝑖 (𝑔)), (𝑖 (𝑔), 𝑔) \in Ω$ with $𝑚 (𝑔, 𝑖 (𝑔)) = 𝑚 (𝑖 (𝑔), 𝑔) = 𝑒$ for all $𝑔 \in 𝐺$ (inverse)

Hence $𝐺$ needn’t be closed under $𝑚$ .

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Local topological group

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