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spaces/S000107/properties/P000229.md

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Since {S107|P86}, it is enough to show that the path-component of $0$
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is {P229}. By {{mathse:5012784}}, this component equals
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$\mathbb{R}^\infty := \{y : y^n = 0\text{ for all but finitely many }n\}$.
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Then it is enough to argue that $\mathbb{R}^\infty$ is contractible ([Explore]([π-Base, Search for `contractible + ~Semilocally simply connected`](https://topology.pi-base.org/spaces?q=contractible+%2B+%7ESemilocally+simply+connected))).
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Then it is enough to argue that $\mathbb{R}^\infty$ is contractible ([Explore](https://topology.pi-base.org/spaces?q=contractible+%2B+%7ESemilocally+simply+connected)).
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This follows once we argue that $F : \mathbb{R}^\infty \times [0, 1] \to \mathbb{R}^\infty$, $(x, t) \mapsto tx$, is continuous.
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By {{mathse:3961052}}, the subspace topology on $\mathbb{R}^\infty$ coincides with the weak topology,

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