Articles of connectedness

Vertex connectivity question

I’m currently learning about vertex connectivity and I’m having a bit of trouble understanding some of the terms/definitions, namely “local connectivity” and “$k$-connected” and “connectivity of $G$” (definition $2$, $3$ and $4$) – I tried finding the vertex connectivity of the following graph $G$: I know that you can also find the vertex connectivity by […]

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 […]

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 […]

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 $j\notin\{i-1,i,i+1\}$? […]

Are components of complement of a set $S$ always close to $S$

Suppose that $X$ is a connected topological space with a connected subset $S$, and let $K$ denote a component of $X\setminus S$. (All sets listed are assumed non-empty, apart from $\emptyset$ of course.) Question. Is there an example of sets, as above, such that $\overline S\cap\overline K=\emptyset$? Remark. If there is such an example, then […]

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} is not homeomorphic to any circle, but I am not sure how to generalize […]

Showing the topologists sine curve is connected (slight variation)

I came across this post, but I wasn’t sure if $\{\sin(1/x): x > 0\} \cup \{(0,y): y\in [-1,1]\}$ was equivalent to $\{\sin(1/x): x > 0\} \cup \{(0,0)\}$. By equivalent, I mean is solving the problem approached the same way? Also, could someone please explain Stefan’s answer in the link posted above? It seems much more […]

Plane less a finite number of points is connected

I’m trying to prove that if you remove a finite number of points from $\Bbb R^2$, you get a connected set. Let $X\subset R^2$ be a finite set. Every open set in the induced topology of $\Bbb R^2 – X$ is of the form $O-X$, where $O$ is open in $\Bbb R^2$. We must show […]

The rational numbers are totally disconnected but not a discrete space?

Any set of the form $\{x, y\}$ is disconnected. Wouldn’t this imply that the rational numbers is a discrete space, since $\{x\}$ and $\{y\}$ are open?

Connecting an almost idempotent complex matrix to a diagonal 1-0 matrix

$\newcommand{\diag}{\operatorname{diag}}$Let $e\in M_n(\mathbb{C})$ be an almost idempotent matrix in the sense that $\|e^2-e\|<\varepsilon<\frac{1}{4}$ where the norm is taken to be the usual operator norm, and also $\|e\|\leq N$ (where $N>1$). I want to connect $e$ to a matrix of the form $\diag(I_k,0_{n-k})$ (if it is possible) via a path $(e_t)$ such that for each $t$, […]