Prove we cannot make a homeomorphism from [0, 1] to ℝ by contradiction.
35569
Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
52798
Finding counterexamples: bijective continuous functions that are not homeomorphisms
123132
Counterexample for a non-measurable function?
78654
An example of a norm which can't be generated by an inner product
157296
Homeomorphism between the Unit Disc and Complex Plane
129623
A shrinking map that is not a contraction no fixed point.
83326
$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
88099
Not $\sigma$-compact set without axiom of choice
149605
Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?
93923
No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
93606
Showing that $C[0,1]$ is not complete
112306
Two manifolds $X$ and $Y$ with two bijective continuous functions from $X$ to $Y$ and $Y$ to $X$, but not homeomorphic
3296
Proof That $\mathbb{R} \setminus \mathbb{Q}$ Is Not an $F_{\sigma}$ Set
143043
When is a local homeomorphism a covering map?
47231
The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive.
52012
Is there exist a homemoorphism between either pair of $(0,1),(0,1],[0,1]$
148187
Question regarding one-point compactification and homeomorphisms