# connectedness's questions - Chinese 1answer

1.710 connectedness questions.

### Can someone help me figure out the proof for this question related to Hamiltonian graphs?

Prove that removal of a subgraph of order k from a Hamiltonian graph of order n(n>=k) will result in no more than k connected components.

### 1 What are the connected components of $\Bbb R^2 \setminus \big(\{(x,y) \in \Bbb R^2, x^2+y^2=1\} \cup \{(x,y)\in \Bbb R^2, y=x^2+1\}\big)$? [on hold]

1 answers, 50 views general-topology connectedness
What are the connected components of $\Bbb R^2 \setminus \big(\{(x,y) \in \Bbb R^2, x^2+y^2=1\} \cup \{(x,y)\in \Bbb R^2, y=x^2+1\}\big)$ ? I made a drawing and to me the connected components are: ...

### -1 What is the method to determine if a space is compact, complete and connected? [on hold]

Let $X=\{(x,y,z)\in \Bbb R ^3 , x^2+y^2+z^2=1\}$ and $Y=\{(x,y,z)\in X,z\neq 0\}$ Are $X$ and $Y$ compact? complete? connected? I know the definitions, but I would like to know if there is a method ...

### -2 simply connected difference in $\mathbb{R}^2$ and $\mathbb{R}^3$ [on hold]

0 answers, 32 views connectedness curves
why $\mathbb{R}^2 - \{(0,0)\}$ is not simply connected but $\mathbb{R}^3 - \{(0,0,0)\}$ is simply connected? I can't understand

### 2 Connected locally compact abelian groups

Is the category $\text{LCA}_c$ of connected locally compact Hausdorff abelian groups an abelian category? My feeling says no, however I can't immediately find a counterexample. Alternatively, I'd ...

### 1 Find a topological space $X$ which is connected but has three path components.

What would a connected topological space $X$ look like with three path components? I know that since it has a finite number of path components, these components are closed but I'm not sure if that ...

### 3 Do partitions of a square into two sets always connect one pair of opposite edges?

0 answers, 44 views general-topology connectedness
Let $S=[0,1]\times [0,1]$ be the closed unit square. Suppose we label its four edges in cyclic order as $E_1,E_2,E_3,E_4$ so that $E_1$ is parallel to $E_3$ and $E_2$ is parallel to $E_4$. Now, ...

### Equivalence relation on topological space such that each equivalence class and the quotient space is connected [duplicate]

Let $X$ be a topological space and $\sim$ be an equivalence relation on $X$ such that the quotient space $X/\sim$ is connected and each equivalence class of $\sim$ is connected (as a subspace of $X$ )....

### Extremally disconnected implies $T_1$?

Let $T =(X, \tau)$ an extremally disconnected topological space, i.e. for every $A \in \tau$, $\overline{A} \in \tau$. Thus, $T$ is also a $T_1$ space? If not, there exists a simple counterexample? ...

### Path-connectedness of the boundary of a set?

Is the boundary of a simply connected set of the plane bounded and with non-empty interior a path-connected set? Can I consider as counterexample the area between the x-axis and the topologist's sine ...

### 2 Čech-Stone compactification of extremally disconnected space

For reference: A space is extremally disconnected iff the closure of every open set is clopen. I am trying to understand a simple proof for the fact that the Čech-Stone compactification $\beta X$ ...

### 3 Topological property of preimage of map $f$ taking the coefficients of characteristic polynomial?

Suppose we define $f\colon \mathcal{M}(n \times n; \mathbb R) \to \mathbb R^n$ by \begin{align*} A \mapsto (\alpha_{n-1}, \dots, \alpha_0), \end{align*} where $(\alpha_{n-1}, \dots, \alpha_0)$ are ...

### -1 Path connectedness of a subset of $R^2$

Let us consider the following subset of $\mathbb{R}^2$. $\cup_{ n=1}^{\infty}\{(x, y) \in \mathbb{R}^2\ | \ x = ny\} \subset \mathbb{R}^2$ How to show that above set is path connected?

### Multiply connected region

0 answers, 25 views complex-analysis connectedness
Consider the region bounded by the Nyquist plot: Why is the region not simply connected? I have trouble understanding if the area bounded by the small elliptical-like shape belongs to the region ...

### 1 Show that $A∪B$ is connected

1 answers, 39 views general-topology connectedness
Define subsets $A,B$ of $\Bbb{R}^2$ by $$A=\{(x,y)∈\Bbb{R}^2 : x^2+y^2=1\}$$ $$B=\{(a(t) \cos t, a(t) \sin t ) \in \Bbb{R}^2 : 1≤t<∞\}$$ where $a(t)=\frac{1-t}{t}$ Show that $X=A∪B$ is ...

### 14 The closure of a connected set in a topological space is connected

2 answers, 7.682 views general-topology connectedness
This problem is from Rudin. I am trying to Prove that the closure of a connected set is always connected. Here is my proof. Let $E$ be a connected set in a space $X$. Suppose to the contrary that the ...

### 2 Weak, Regular, and Strong connectivity in directed graphs

3 answers, 1.118 views graph-theory algorithms connectedness
There are 3 types of connectivity when talking about a directed graph $G$. 1) weakly connected - replacing all of $G$'s directed edges with undirected edges produces a connected (undirected) graph. ...

### 1 Prob. 11, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: Any example of a connected subspace with disconnected interior and boundary?

1 answers, 36 views general-topology connectedness