I'm learning real analysis. Open ball: The collection of points $x \in X$ satisfying $|x - x_{0}| < r$ is called the open ball of radius $r$ centered at $x_{0}$ Neighborhood: A neighborhood of $x_{0} \in X$ is an open ball of radius r > 0 in $X$ that is centered at $x_{0}$ I'm usin...

In 1974, a paper titled On the Classification of Knots appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the other: From that point on, $10_{161}$ and $10_{162}$ became kn...

Prove that the straight lines whose direction cosines are given by the relations $al+bm+cn=0$ and $fmn+gnl+hlm=0$ are parallel if $\sqrt {af} \pm \sqrt {bg} \pm \sqrt {ch}=0$. My Attempt: Given: $$al+bm+cn=0..........(1)$$ $$fmn+gnl+hlm=0........(2)$$ From equation $(1)$, $$l=-\dfrac {bm+cn}{...

Problem If $\{ B_\lambda : \lambda\in\Delta\}$ is any collection of open subsets of $X$ whose union is $X$, a function on $X$ is continuous iff its restriction to each $B_\lambda$ is continuous. Context This is not true for general covers. For example, $X=(0,2)$ is the union of $(0,1)$ a...