2
votes
1answer
57 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
0
votes
1answer
33 views

How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
0
votes
0answers
30 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
0
votes
1answer
34 views

Compact and connected

Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by $$\{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...
2
votes
1answer
31 views

Compact, extremely disconnected spaces of weight $\omega_1$

Weight of a topological space is the minimal cardinality of a basis of the topology. A space $X$ is extremely disconnected if open sets in $X$ have open closures. Is there an example in ZFC of a ...
1
vote
1answer
54 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
1
vote
1answer
48 views

Constructing a Set with Connected Interior

Suppose that $K\subset\mathbb C$ is a compact set with non-empty interior and suppose that $a\in\operatorname{int} K$. I want to construct a set $M$ with the following properties: $M\subseteq K$; ...
1
vote
1answer
48 views

Compactness and connectedness of the topological space?

Let $X=\mathbb N$ be equipped with the topology generated by the basis consisting of sets $A_n = \{n,n+1,n+2,\ldots\} ,n \in \mathbb N $ . Then $X$ is compact and connected Hausdorff and connected ...
2
votes
1answer
112 views

Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
5
votes
2answers
55 views

Connectedness and compactness of a union of two sets

Let: $$A=\Big\{ (x,y) \in \mathbb R^2: 0 \le x \le 1, y=\frac{x-1}{n},\, n\in \mathbb N \Big\}$$ $$B=\Big\{ (x,y) \in \Bbb R^2: 0 \le x \le 1, y=\frac{x}{n},\, n\in \mathbb N \Big\}$$ Is $A \cup B$ ...
3
votes
2answers
143 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
4
votes
1answer
54 views

Example of topological space where pseudo-component differ with intersection of clopen sets.

It is well known fact that connected component $C_x$ of a point $x$ from some topological space $\tau$ is contained in every clopen set containing $x$ (so it's intersection $M$ also contains $C_x$). ...
1
vote
2answers
39 views

Negative exponential distance

Let $X := \left\{(a_k)_{k \in \mathbb N}, a_k \in \mathbb C\right\}$. Let $d\left( (a_k)_{k \in \mathbb N}, (b_k)_{k \in \mathbb N} \right) := e^{-u}$ with $u$ the smallest integer $k$ such that $a_k ...
2
votes
1answer
231 views

Quasicomponents and components in compact Hausdorff space

Let $X$ be a compact Hausdorff space, $x,y\in X$ and $\mathcal{A}$ a colection of closed subspaces of $X$ such that for every $A\in \mathcal{A}$ then $x$ and $y$ are in the same quasicomponent of $A$. ...
2
votes
1answer
146 views

Can the complement of a simply connected set in $\bar{\mathbb{C}}$ in an open set always be covered by a simply connected union of balls?

I believe the following to be true, but am worried my intuition does not account for fractally things: Let $K\subset\bar{\mathbb{C}}$ ($\bar{\mathbb{C}}$ being the Riemann sphere) be closed (thus ...
3
votes
2answers
288 views

Can $X$ have compact connected components?

Let $X$ be a Hausdorff, locally compact but non-compact topological space. If the (Alexandroff) one-point compactification is connected, can $X$ have compact connected components?
2
votes
1answer
55 views

Function for which taking preimages preserves limit points

Suppose we have a surjection $f : X \to Y$ between topological spaces. What is the weakest assumption on $f$, $X$ and $Y$ you can think of that endows $f$ with the following property? If $A ...
1
vote
0answers
80 views

Proofs that quasicomponents of compact Hausdorff spaces are connected

Nuno's answer to Any two points in a Stone space can be disconnected by clopen sets uses (and proves) the following: Theorem: Let $X$ be a compact Hausdorff space. Then the quasicomponents of $X$ are ...
1
vote
1answer
137 views

Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
2
votes
3answers
97 views

What about the compactness and connectedness of $[0,1]$ in this topology?

If the set $\mathbf{R}$ of all real numbers has the topology consisting of all sets $A$ such that $\mathbf{R} \setminus A$ is either countable or all of $\mathbf{R}$. What can we say about compactness ...
0
votes
1answer
125 views

Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected. [duplicate]

Let $X$ be a Hausdorff space and let $C_0 \supset C_1 \supset ...$ be a decreasing sequence of compact connected subsets of $X$. Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected. ...
2
votes
1answer
267 views

Is this set compact? Connected?

Is this set compact? Connected? $S=\{(x,y,z)\in\mathbb{R}^3:z=x^2+y^2+1\}$ for $z\le 1$ this set is not defined, but for $z>1$ we are getting circles! I imagined this as a bunch of ...
-3
votes
1answer
95 views

Show that $X$ is locally compact, and each connected component of $X$ is a point.

Let $X$ be a Hausdorff space and suppose that each point has a neighborhood basis of simultaneously open and compact neighborhoods. Show that $X$ is locally compact, and each connected component of ...
3
votes
3answers
145 views

Show $X=\left\{x \in [0,1]: x \neq \frac1n\text{ for any }n \in \Bbb N\right\}$ is neither compact nor connected

I am stuck on the following question: Let $X=\{x \in [0,1]: x \neq \frac1n: n \in \Bbb N\}$ be given the subspace topology. Then I have to prove that $X$ is neither compact nor connected. Can ...
1
vote
1answer
105 views

Prove that $ \bigcap _{i=1}X_i$ is connected. [duplicate]

Let $X$ be a compact Hausdorff space and let $X_1 \subset X_2 \subset X_3 \subset \cdots$ be a sequence of closed, connected subspaces. Prove that $\bigcap_{i=1}^\infty X_i$ is connected. Give an ...
3
votes
0answers
39 views

Strongly additive open cover

Call an open cover $\mathscr{U}$ of a metric space $M$ strongly additive if whenever $U,V\in\mathscr{U}$ and $U\cap V\ne\emptyset$, then $U\cup V\in\mathscr{U}$. Prove that $M$ is compact and ...
4
votes
0answers
119 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
2
votes
1answer
56 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
1
vote
1answer
106 views

Compactness and connectedness on $M_n(\mathbb R)$

Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices. Which of the following are compact and which are connected? a) The set of all invertible matrices b) The set of all orthogonal ...
3
votes
2answers
154 views

Imposing the topology of open rays in $\Bbb R$

After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing ...
4
votes
1answer
121 views

Minimal Connected Set containing a Closed Connected Set in a Compact Space

This question came from Dugundji's $\textit{Topology}$: Given a compact, connected space $X$, let $A \subset X$ be closed. Prove that there exists a closed, connected set $B \subset X$ such that $A ...
3
votes
1answer
180 views

Topological properties of symmetric positive definite matrices

Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? (a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$. (b) $S$ is ...
5
votes
2answers
206 views

Connectedness of $\beta \mathbb{R}$

I am not very familiar with Stone-Čech compactification, but I would like to understand why the remainder $\beta\mathbb{R}\backslash\mathbb{R}$ has exactly two connected components.
13
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
2
votes
0answers
146 views

Connectedness of the complement of a compact “small” subset of $\mathbb R^n$

Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le ...