Articles of general topology

proof of connectedness of $\Bbb{R}$

I’m trying to give a proof of the connectedness of $\Bbb{R}$ (with the standard topology): For the sake of argument, suppose $\Bbb{R}$ is not connected. Let $U\subset\Bbb{R}$ be open, closed, nonempty, and $U\not=\Bbb{R}$. Pick $a\in U$ and $b\in \Bbb{R}\setminus U$. Assume WLOG $a<b$. Set $A:=U\cap[a,b]$. Note that $A$ is nonempty since $a\in A$ and bounded. […]

How to show the covering space of an orientable manifold is orientable

I’m trying to prove this using purely topological arguments, no differential geometry as I haven’t been exposed to it. I’ve been playing around with definitions a bit and here’s what I have so far. Let $M$ be an orientable manifold. Let $N$ be its covering space. Then we have an orientation function $\mu : M […]

Another example of a connected but non path connected set

I’m looking at the polar graph of the function $f:[\frac\pi2,\infty)\to \Bbb{R}^2$ defined by $\ r=f(\theta)=e^{\frac 1 \theta}$, the graph of this is the set of points of the form $(e^{\frac 1 \theta}\cos(\theta),e^{\frac 1 \theta}\sin(\theta))$. That set(which happens to be the polar graph) is path-connected since it’s the image of a path-connected set by a continuous […]

The empty set is a neighborhood?

The following axioms of a Topological space is from Wikipedia: Neighbourhoods definition This axiomatization is due to Felix Hausdorff. Let $X$ be a set; the elements of $X$ are usually called points, though they can be any mathematical object. We allow $X$ to be empty. Let $N$ be a function assigning to each $x$ (point) […]

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) = \sum_{i=1}^{\infty} |\xi_i – \eta_i|, \qquad x = (\xi_i), y = (\eta_i). $$ Theorem: A subset $M$ of $l_1$ is totally bounded (pre-compact) […]

Prove Intersection of Two compact sets is compact using open cover?

Let A and B be compact subset of R To show intersection of A and B is compact, I need to show that for any open cover for intersection has finite subcover. It is quite straightforward for Union of two compact sets, but how can I start with the intersection casE?

Alexander Polynomial of the stevedore knot using Fox's free calculus

I want to find the Alexander Polynomial of the stevedore knot using the free calculus of Fox. A presentation for the knot group of the stevedore knot is $$G=(x,a:xa^3xa^{-2}x^{-1}=a^2xa^{-2})$$ We can rewrite the relator as $r=a^2x^{-1}a^{-2}xa^3xa^{-2}x^{-1}=1$. To find the Alexander matrix we find the free derivatives of the relator with respect to each generator: $$\frac{\partial […]

Understanding the definition of continuity between metric spaces.

As we know that $\epsilon-\delta$ definition of continuity between metric spaces $X$ and $Y$ can be stated as follows: A map f:$(X, d_X)\rightarrow (Y, d_Y)$ is said to be continuous at a point $p\in$ X if for a given $\epsilon >0$, $\exists$ $\delta >0$ such that $d_X(p,x)< \delta\Rightarrow d_Y(f(p),f(x))< \epsilon $ I need help to […]

Is the pullback of a *not necessarily continuous* open map along a continuous map open?

The pullback of an open map in Top is open. We could consider more generally the pullback in Set along a continuous function $g : A \to B$ of a function $f: C \to B$ which take opens to opens, but is not necessarily continuous. Does this new function $pr_1: A \times_B C \to A$ […]

What is the relation between the usual topology of $S^1$ and its subspace topology in Homeo$(S^{n+1})$?

Let the set of all self homeomorphisms of $S^{2n+1}$ – $\operatorname{Homeo}(S^{2n+1})$, be given the compact open topology. Fix $a_0,\cdots,a_n\in\mathbb Z$ to be $n+1$ coprime integers. Let $S^1$ act on $S^{2n+1}$ as follows – $$\lambda\cdot(z_0,\cdots,z_n)=(\lambda^{a_0}z_0,\cdots,\lambda^{a_n}z_n)$$ $($The resulting quotient space is what is known as the weighted projective space $W\mathbb P(a_0,\cdots,a_n))$ The action is clearly faithful and […]