**1.710 connectedness questions.**

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.

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:
...

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 ...

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

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 ...

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 ...

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, ...

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 $X$ )....

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?
...

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 ...

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 $\beta X$ ...

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 ...

Suppose $f\colon \mathbb R ^2 \to \mathbb R$ is defined by $(x_1, x_2) \mapsto x_1+x_2 + x_1x_2$. Let $E \subset \mathbb R$ be a connected set containing $0$ and $E$ is not the singleton $0$. Is $f^{-...

I am extracting information of my problem from proving an application problem, if there is anything not clear, please let me know.
Suppose we parameterize a structured matrix in following form
\begin{...

I have to prove that if $f:S^{n} \to \mathbb{R} $ (where $S^{n}= \{(x_1, ..., x_{n+1})\in\mathbb{R}^{n+1} | x_1^2+...+x_{n+1}^2=1\} $ is continuous, then is not injective. If possible, I would like ...

Let $V$ be the span of $(1,1,1)$ and $(0,1,1) \in \Bbb R^3$. Let $u_1=(0,0,1),u_2=(1,1,0)$ and $u_3=(1,0,1)$. Which of the following is/are correct?
$1. \ $ $(\Bbb R^3 \setminus V) \cup \{(0,0,0)...

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?

If we define a metric $d(x,y)=\begin{cases}|x|+|y|,&\text{if }x\neq y\\0,&\text{if }x=y\end{cases}$ on $\mathbb{R}$, which of following options are true?
Every compact set is finite,
$\mathbb{...

Suppose we have a monic polynomial with real coefficients
\begin{align*}
p(x) = x^n + \alpha_{n-1} x^{n-1} + \dots + \alpha_1 x + \alpha_0.
\end{align*}
Let $M > 0$ be some fixed real number and ...

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N\neq 0$. Let
\begin{align*}
E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-...

Consider the following situation: Let $G$ be a topological group with identity element $e \in G$ and let $G_0$ denote the identity component of $G$, i.e. the connected component of $e$ (actually, the ...

Consider $\Bbb R_n$ with the usual topology for $ n = 1,2, 3$.
Each of the following options gives
topological spaces $X$ and $Y$ with respective induced topologies.
In which option is $X$ ...

Given $A(t) \in M_n$, where $t \in \mathbb{R}$.
$A(t)$ is smooth and continuous.
Given $f(A(t)) \in \mathbb{C}$
$f(A(t))$ is smooth and continuous. f may contain inverses, transposes, determinants ...

Let $C \subseteq \mathbb{R}$ be a set that has uncountably many connected components. What can be said about the openness/ closedness of $C$ in $\mathbb{R}$?
(One such example is the Cantor set. It ...

Let $X=[0,+ \infty[ $ and $d(x,y)=|\frac{1}{1+x^2}- \frac{1}{1+y^2}| $
1) Show that $(X,d)$ and $(]0,1], d_{2})$ are homeomorphic (where $d_{2}=|x-y|$)
2) Is the space $(X,d)$ connected? compact?...

This is something of a pedagogical question. Most of us have scratched our heads at some point as to why the definition of "$f$ is continuous" requires that the preimage of any open set under $f$ is ...

I'm looking for a topological space in which every point has a neighbourhood basis of connected sets but is not locally connected. Note that locally connected means that every point has a ...

Is $A=\{(x,y): x^2+y^2=1\}$
is connected in $ℝ^2$?
From its graph, I would conclude that it's not path connected.

Suppose $X$ is metric, compact, connected, and $p\in X$.
An arc is a copy of $[0,1]$.
Is it possible that every two points in $X\setminus \{p\}$ can be joined by an arc, but there is no arc in $X$ ...

I'm working on a problem, which I've managed to formulate in two different ways. I'll explain both of them below. To be frank, I'm not sure this question is totally well-defined, but I hope that ...

Fix a product of projective spaces (say over R) $X=\mathbb{P}^{n_1}\times \ldots \times \mathbb{P}^{n_k}$ and a polynomial $f$ which is homogeneous of degree $d_i$ in the coordinates of the i-th space....

I would like to prove that every open connected subset of $\mathbb{R}^n$ is path connected.
Let us choose $E$ to be a such open connected subset, then given any point $p \in E$, we will define $F$ be ...

The Anti-de Sitter space is $AdS_n=\frac{O(2,n-1)}{O(1,n-1)}$, as a homogeneous symmetric space.
Is the space connected or not, especially for $n=2$?
Is there a general method to judge that?

I was working with this Theorem and somehow I don't see clearly how it's being proved.
Theorem:
Let $(X, \tau)$ be topological space and $A_\alpha \subset X$ be connected for every $\alpha \in I$, ...

How would this graph be classified? And what properties does it have? What would be a natural thing to study with this graph? I know it has 40 nodes. It's undirected. It's planar if I'm not mistaken. ...

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in Steen-...

The Question:
Define the subset of $\Bbb R^3$
$$S=\{ (x,y,z) \in \Bbb R^3: x^2+y^2+z^2=1 \}$$
Suppose that $f:S \rightarrow \Bbb R$ is a continuous function. Prove that there exists $p \in S$ such ...

In Munkres - Topology, in section 24, question 8a), we are asked "Is the product of path connected spaces necessarily path connected?" Here is my proof:
In the product topology, yes; since for any ...

Problem:
Suppose that you can write a set $A \subset \Bbb R^n$ as $A \subset F_1 \cup F_2$ where $F_1, F_2$ are closed, and
$A \cap F_1 \cap F_2=\emptyset $
$F_1 \cap A \neq \emptyset$
$F_2 \cap ...

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 ...

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 ...

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 ...

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.
...

Here is Prob. 11, Sec. 24, in the book Topology by James R. Munkres, 2nd edition:
If $A$ is a connected subspace of (a topological space) $X$, does it follows that $\mathrm{Int} A$ and $\mathrm{Bd}...

Let $\{X,\tau\}$ be a topological space, $E\subset X$; prove that $cl(E)$ is connected if and only if $E$ is not union of two non-empty subsets $A$ and $B$ such that $cl(A)$∩$cl(B)$= $\emptyset$
when ...

Let V be span of $(1,1,1) $and $(0,1,1)$.Let $p=(0,0,1),q=(1,1,0),r=(1,0,1)$
Then prove that $ \Bbb R ^3-V \cup$ $\{tp+(1-t)r:t \in [0,1]\}$ is not connected.
$V=\{a(1,1,1)+b(0,1,1):a,b \in \Bbb R\}$ ...

How to determine if a simple graph of $7$ vertices where the degree of each vertex is at least $3$ is connected?
According to this answer we need to check that $|E|>{7-1\choose 2}=15$. If the ...

Let $(X,\tau)$ be a topological space and $E,Y\subset X$ such that $E$ is connected, $E\cap Y\neq\emptyset$ and $E\cap(X\setminus Y) \neq \emptyset$. Thus $E\cap\partial Y \neq \emptyset$. Any help is ...

Let $C \subset \mathbb R^n$ a connected set. Show that if $x$ is a limit point of $C$, then $C \cup {x}$ is connected.
I first assumed that $S = C \cup {x}$ is disconnected, then, by definition, ...

Let $(X_i)_{i \in I}$ be a family non empty topological spaces. Prove that $\coprod_{i \in I} X_i$ is totally disconnected if $X_i$ is totally disconnected for all $i \in I$
My book gives the hint:
...

- general-topology
- real-analysis
- metric-spaces
- compactness
- graph-theory
- continuity
- path-connected
- proof-verification
- algebraic-topology
- analysis
- complex-analysis
- examples-counterexamples
- matrices
- proof-explanation
- topological-groups
- combinatorics
- calculus
- multivariable-calculus
- elementary-set-theory
- separation-axioms
- linear-algebra
- lie-groups
- homotopy-theory
- covering-spaces
- proof-writing

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