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- wikipedia: Fréchet–Urysohn_space
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name: Fréchet–Urysohn space on Wikipedia
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---
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$X$ is the subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$, with the final
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topology with respect to the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$,
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$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$. By definition of final topology, this means that $\mathbb{R}^\infty$ has the finest topology such that each such inclusion map is continuous.
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Equivalently, the set $U \subset \mathbb{R}^\infty$ is open if and only if $U \cap \mathbb{R}^n$
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is open in $\mathbb{R}^n$ for each $n$, where we identify each Euclidean space $\mathbb{R}^n$ with
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its image.
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$X$ is the subset $\mathbb{R}^\infty$ of eventually $0$ sequences in $\mathbb{R}^\omega$,
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with the [final topology](https://en.wikipedia.org/wiki/Final_topology)
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with respect to the standard inclusion maps $\mathbb{R}^n \hookrightarrow \mathbb{R}^\infty$,
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$x \mapsto (x^1, \ldots, x^n, 0, \ldots)$.
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Thus, a set $U \subseteq \mathbb{R}^\infty$ is open iff $U \cap \mathbb{R}^n$
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is open in $\mathbb{R}^n$ for each $n$,
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where we identify each Euclidean space $\mathbb{R}^n$ with its image.
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Equivalently, $\mathbb{R}^\infty$ is the direct limit $\varinjlim \mathbb{R}^n := (\bigsqcup_{i = 1}^\infty \mathbb{R}^i)/\sim$ of the directed
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system consisting of Euclidean spaces and standard inclusion maps
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