Timeline for Characterization of Unusual Topologies of $\mathbb R$

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May 8, 2014 at 3:10	history	edited	François G. Dorais		edited tags
Jun 4, 2011 at 7:19	comment	added	KP Hart		In reply to the ``Note that ...'': there is no `typical' bijection between $[0,1]$ and $(0,1)$. All that happens is that $0$ and $1$ disappear along the sequences of iterates $f^n(0)$ and $f^(1)$; these can be as wild as you want (dense, somewhere dense but not everywhere, dense in the Cantor set, ...) and outside those sequences you can mess up the structure of the interval even more.
Jun 1, 2011 at 20:47	history	edited	Asaf Karagila♦	CC BY-SA 3.0	Added more constraints and edited.
Jun 1, 2011 at 20:35	comment	added	Asaf Karagila♦		@David Speyer: This is a good point, I'll restate my question.
Jun 1, 2011 at 20:32	comment	added	David E Speyer		Do you want to impose some sort of compatibility between this topology and the standard one? Otherwise, you are just asking: What all are the topological spaces $X$, with cardinality the continuum, such that there is a continuous bijection $X \to X \setminus \{ x,y \}$ . There are going to be tons of these. For example, take any topological space $Y$ of continuum cardinality and take its disjoint union with a countable number of isolated points.
Jun 1, 2011 at 20:24	history	asked	Asaf Karagila♦	CC BY-SA 3.0