Timeline for Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$? [closed]

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when toggle format	what		by	license	comment
Nov 23, 2017 at 17:54	history	closed	YCor Neil Strickland Jan-Christoph Schlage-Puchta user1073 Ramiro de la Vega		Not suitable for this site
Nov 23, 2017 at 13:06	comment	added	YCor		Additional exercise: there's a countable subset of $\mathrm{Cont}(\mathbf{R},\mathbf{R})$ that is dense in $\mathbf{R}^\mathbf{R}$.
Nov 23, 2017 at 12:37	comment	added	YCor		"this argument" is indeed not well-written: it should be "a non-Borel self-map of $\mathbf{R}$ is not the pointwise limit of a sequence of continuous functions". This example shows that one has to be careful with dealing with limits in non-metrizable spaces, such as, typically, uncountable products.
Nov 23, 2017 at 11:25	history	edited	Martin Sleziak	CC BY-SA 3.0	typo
Nov 23, 2017 at 10:43	review	Close votes
Nov 23, 2017 at 17:54
Nov 23, 2017 at 10:24	vote	accept	Dominic van der Zypen
Nov 23, 2017 at 10:18	vote	accept	Dominic van der Zypen
Nov 23, 2017 at 10:24
Nov 23, 2017 at 10:05	comment	added	Jochen Glueck		@Phil-W I think, this argument only shows that $\operatorname{Cont}(\mathbb{R},\mathbb{R})$ is not sequentially dense in $\mathbb{R}^{\mathbb{R}}$.
Nov 23, 2017 at 10:05	answer	added	Taras Banakh		timeline score: 14
Nov 23, 2017 at 10:03	answer	added	Jochen Glueck		timeline score: 10
Nov 23, 2017 at 9:54	comment	added	Dominic van der Zypen		If you can elaborate on this a bit, you can post it as an answer and we can close this thread.
Nov 23, 2017 at 9:51	comment	added	Phil-W		I would say no, since a non Borel self-map of $\mathbb{R}$ is not the limit by pointwise convergence of continuous functions.
Nov 23, 2017 at 9:33	history	asked	Dominic van der Zypen	CC BY-SA 3.0