Articles of general topology

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 […]

Prove that $\partial A$ is a cutset of connected $X$ if $\operatorname{Int}(A)$ and $\operatorname{Int}(X – A)$ are nonempty

Exercise 6.23 (p.202) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa asks: Let $X$ be a connected topological space and $A$ be a subset of $X$. Prove that if $\operatorname{Int}(A)$ and $\operatorname{Int}(X – A)$ are nonempty, then $\partial A$ is a cutset, and the pair of sets, $\operatorname{Int}(A)$ and $\operatorname{Int}(X […]

$A$ closed in $Y$ and $Y$ closed in $X$ , then $A$ is closed in $X$

This sounds really simple and I’m struggling with it. I first tried to show that $X-A$ had to be closed by trying to show the complementary had to be open (trying to express it as union or intersection of known opens), but I couldn’t do it: $(X-A)$ has to be open, and that equals $(X-Y)\cup […]

How well connected can a (special) partition of $\Bbb R^2$ be?

Let $\{A_i\}_{i\in I}$ be a family of subsets of $\Bbb R^2$ (where $I=\Bbb N$ or $\Bbb Z$; I don’t know if it makes a difference) such that $\bigcup_{i\in I} A_i=\Bbb R^2$ $i\ne j\implies A_i\cap A_j=\emptyset$ $A_i\ne\emptyset$ $A_i$ is connected $A_i\cup A_{i+1}$ is connected How often can it happen that $A_i\cup A_j$ is connected for $j\notin\{i-1,i,i+1\}$? […]

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a compact $G_\delta$ with respect to $F$. Clearly $K$ is compact as a subset of $X$. Must it […]