Answer by math is love for Non-homeomorphic spaces that have continuous...
What about $X= \mathbb R$ with usual topology and $Y=[a,b]$ with subspace topology?Then $(a,b)$ is a subspace of $[a,b]$ which is homeomorphic to $\mathbb R$ and $[a,b]$ is a subspace of $\mathbb R$...
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Answer by Jens Reinhold for Non-homeomorphic spaces that have continuous...
I know this is super old, but somebody asked the same question again (Non-homeomorphic topological spaces) and so I wanted to share a "proof by picture" that settles the question. (I came up with this...
View ArticleAnswer by Will Brian for Non-homeomorphic spaces that have continuous...
I just posted an answer to this related MO question. To sum up the part that's relevant here:Let $\mathcal N$ be the Baire space and let $X$ be any other zero-dimensional Polish space that is not...
View ArticleAnswer by Maarten Derickx for Non-homeomorphic spaces that have continuous...
I've asked myself this question some time ago and found some counterexamples. After having done this I asked myself what the "smallest" counter example would be. First of all notice that if the set of...
View ArticleAnswer by Michał Kukieła for Non-homeomorphic spaces that have continuous...
You may find the paper "Bijectively related spaces. I. Manifolds" by P. H. Doyle and J. G. Hocking intersting. They cite some related work, which is also worth checking.There is a later paper "Unusual...
View ArticleAnswer by ethan akin for Non-homeomorphic spaces that have continuous...
I don't have my copy of Kelley handy but I think in chapter 1 he gives the example where X is a countable disjoint union of open intervals and a countable discrete set while Y is a countable disjoint...
View ArticleAnswer by O.R. for Non-homeomorphic spaces that have continuous bijections...
Here is an example which comes from using the spaces of Charles Siegel's post. One have a continuous bijection from [0,1) to the circle given by theexponential function ( t-->exp(2ipit) ). The idea...
View ArticleAnswer by S. Carnahan for Non-homeomorphic spaces that have continuous...
Here's a continuum analogue of Gerhard Paseman's answer: Let $X$ and $Y$ be topological spaces whose underlying sets are $\mathbb{R}$. As topological spaces, $X$ is the disjoint union of the open...
View ArticleAnswer by Gerhard Paseman for Non-homeomorphic spaces that have continuous...
Recycling an old (ca. 1998) sci.math post:" Anyone know an example of two topological spaces $X$ and $Y$ with continuous bijections $f:X\to Y$ and $g:Y\to X$ such that $f$ and $g$ are not...
View ArticleAnswer by Charles Siegel for Non-homeomorphic spaces that have continuous...
My favorite, which is on the wikipedia page for "homeomorphism", is $\phi:[0,2\pi)\to S^1$, by $\phi(\theta)=(\cos\theta,\sin\theta)$, which is continuous and bijective, but not a homeomorphism.
View ArticleNon-homeomorphic spaces that have continuous bijections between them
What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
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