query: Show $f(x)=\frac{x}{1-|x|}$ is a homeomorphism from (-1,1) to $\mathbb{R}$
positive_passages.docid: 157268
positive_passages.text: Proving a bijection with absolute value
negative_passages.docid: 46771
negative_passages.text: $X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)
negative_passages.docid: 140052
negative_passages.text: Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
negative_passages.docid: 40209
negative_passages.text: $f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
negative_passages.docid: 67020
negative_passages.text: Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$
negative_passages.docid: 141643
negative_passages.text: Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
negative_passages.docid: 157296
negative_passages.text: Homeomorphism between the Unit Disc and Complex Plane
negative_passages.docid: 55902
negative_passages.text: Show $R[x]/(x)\cong R$
negative_passages.docid: 90649
negative_passages.text: Consider $f(x)=mx$ and $g(x) =bx$ in which $m>0$ and $b>0$. Does there exist any homeomorphism $h$ such that $h \circ f$ = $g \circ h$
negative_passages.docid: 60485
negative_passages.text: Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval $(0,1)$
negative_passages.docid: 76610
negative_passages.text: Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
negative_passages.docid: 82458
negative_passages.text: Homeomorphism between the unit disc and the unit square
negative_passages.docid: 101778
negative_passages.text: Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
negative_passages.docid: 144246
negative_passages.text: Subset of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$
negative_passages.docid: 81264
negative_passages.text: Is $[0, 1) \times (0, 1)$ homeomorphic to $(0, 1) × (0, 1)$?
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 15312
negative_passages.text: Show that $R \cap R^{−1}$ is an equivalence relation on $X$?