Articles of connectedness

Consider the “infinite broom”

Consider the “infinite broom” $X$ pictured in figure below. Show that $X$ is not locally connected at $p$, but is weakly connected at $p$.[Hint: Any connected neighborhood of $p$ must contain all the points $a_i$] For simplicity, I took, $X\subset \mathbb{R}^2$, such that $p=0\times 0$ and $a_1=1\times 0$, so that consider the subspace topology of […]

Are there path-connected but not polygonal-connected sets?

This question came up in my mind. In the scope of normed spaces, does there exist a path-connected but not polygonal-connected set? I’d rather say no for open sets (my intuition is that they’re ‘locally convex’). Thanks for your suggestions.

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I’ll first define what I mean by a ‘component of a topological space’: For a topological space $X$, write $x\sim y$ if $\exists\ Y \subset X$ such that $Y$ is connected and $x, y \in Y$ (this is an equivalence relation. […]

The complement of every countable set in the plane is path connected

This question already has an answer here: Arcwise connected part of $\mathbb R^2$ 3 answers If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected? [duplicate] 1 answer Prove that the complement of $\mathbb{Q} \times \mathbb{Q}$. in the plane $\mathbb{R}^2$ is connected. [duplicate] 5 answers

How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?

This question already has an answer here: Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate] 2 answers $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected? 3 answers

Homotopy equivalence induces bijection between path components

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. Let $f:X\to Y$ be a homotopy equivalence. We define $G:C(X)\to C(Y)$ by $G(C(x))=C(f(x))$. Then I want to prove that: […]

A binary sequence graph

Define a graph $H(n, 2)$ as follows. Each vertex corresponds to a length $n$ binary sequence and two vertices are adjacent if and only if they differ in exactly two positions. I want to find three things. (1) The number of vertices (2) The degree of each vertex in this graph (3) The number of […]

$W$ be an $m$-dimensional linear subspace of $\mathbb R^n$ such that $m\le n-2$ , then is it true that $\mathbb R^n \setminus W$ is connected?

Let $W$ be an $m$-dimensional linear subspace of $\mathbb R^n$ such that $m\le n-2$ , then is it true that $\mathbb R^n \setminus W$ is connected (hence path-connected as it is open in $\mathbb R^n$ ) ? My motivation comes from what I can intuitively feel is that if we remove a straight line from […]

On the Riemann mapping theorem

Let’s take the family of analytic one to one functions, $f:G\to \mathbb{C}$ (with $G\neq \mathbb{C}$ a region and $z_0\in G$ a fixed point) such that $|f|<1$, $f(z_0)=0$ and $f'(z_0)$ is a real positive number. One question is to find all the regions $G\neq \mathbb{C}$ such that the previous family is non empty. Clearly, thanks to […]

k-Cells are Connected

I am studying real analysis from Baby Rudin, and while the book proves that real intervals are connected, it does not say anything regarding k-cells. I would expect them to also be connected, but do not now how to prove it. I was thinking that if I were able to prove that: $A\subset\mathbb{R^n}$ connected , […]