2
votes
2answers
54 views

Is every Extremally Disconnected Hausdorff Space Regular?

I was wondering, is it true that a Extremally Disconnected Hausdorff Space is Regular? Let $A$ be an open set. I must find a open set $V$ such that $\bar V \subset A$. Since $\bar V$ will be open I ...
1
vote
1answer
46 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
61 views

How to show that $\mathbb{Q}_p^*$ is totally disconnected?

Let $\mathbb{Q}_p$ be the field of p-adic numbers and $\mathbb{Q}_p^*$ the set of invertible elements in $\mathbb{Q}_p$. How to show that $\mathbb{Q}_p^*$ is totally disconnected? Thank you very ...
1
vote
2answers
83 views

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

Prove $(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ is path connected. I know I need to let $(x_0, y_0), (x_1, y_1) \in$$(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ and then consider ...
2
votes
3answers
59 views

Is $\mathbb{Q}^2$ connected?

Is $(\mathbb Q \times \mathbb Q)$ connected? I am assuming it isn't because $\mathbb Q$ is disconnected. There is no interval that doesn't contain infinitely many rationals and irrationals. But ...
0
votes
2answers
55 views

Prove Simply Connected

If $X = U \cup V$ with $U,V$ open and simply connected and $U \cap V$ is path connected, why is $X$ simply connected?
3
votes
3answers
43 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
0
votes
1answer
59 views

Question on Connectedness - Topology by Munkres $23.12$

Question is : Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected. What i have tried is : Suppose $Y\cup A$ has ...
1
vote
1answer
28 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: $M\subseteq K$; ...
2
votes
0answers
40 views

Cofinite topology on an infinite set $X$ is connected?

Here is my proof of that the cofinite topology on an infinite set $X$ is connected. $X$ is connected $\iff$ There are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$. ...
1
vote
1answer
48 views

Possibility of Union of two connected sets being connected with empty Intersection

I know that : If $A$ and $B$ are connected then $A\cup B$ is also connected if $A\cap B\neq \emptyset$ My Question is.. Are there any examples where $A$ and $B$ are connected and $A\cup B$ ...
0
votes
1answer
46 views

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 ...
1
vote
2answers
67 views

Theorem 23.5 from topology by James Munkres

I'm reading the following theorem: I don't understand what would go wrong if you would leave out the part that I marked green. I would think you could safely leave it out and replace the second to ...
1
vote
0answers
50 views

Prove $Y$ is connected

Let $A$ be a connected subspace of $X$ and suppose $A\subseteq Y\subseteq\overline{A}$. Prove that $Y$ is connected. My attempt: Suppose that $Y$ is not connected. Then $Y=U_1\cup U_2$ where $U_1$ ...
3
votes
1answer
53 views

Path-Connectedness of Union

Can I have a hint to prove that $A \cup B$ is not path-connected, where $A = \{(x,y):0 \le x \le 1, y = x/n \text{ for n} \in \mathbb{N} \}$ and $B = \{(x,y):1/2 \le x \le 1, y = 0 \}$?
1
vote
2answers
91 views

Connected set on topology - Contradiction

I have a major problem in understanding connected set! Consider the set $X = (0,1]$. Now lets describe the following sets which can be considered as the basis for the topology on set $X$. 1) For all ...
3
votes
2answers
100 views

Find a subset of $\mathbb{R}^2$ that is path connected but is locally connected at none of its points.

I'm doing this exercise: Let $X$ denote the rational points of the interval $[0,1]\times\{0\}$ of $\mathbb R^2$. Let $T$ denote the union of all line segments joining the point $p=(0,1)$ to points ...
0
votes
2answers
42 views

A contractible space is path connected.

Let the space be $X$ and $\rm{id} \simeq x_0$ where $\rm{id}$ is the identity map on $X$ and $\rm x_0$ is some fixed point in $X$. How do I show that for any two points $a,b \in X$ there is a ...
1
vote
1answer
76 views

Show that there exists a path of class $C^1$ between $x$ and $y$

Let $H$ be an arc connectedness (a subset of $R^n$) , and both $x$ and $y$ components of $H$ Show that there exists a path of class $C^1$ between $x$ and $y$ I feel it's not true, if we ...
0
votes
1answer
31 views

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} ...
6
votes
1answer
82 views

Are graphs of derivatives connected?

One of the basis results in real analysis is the Darboux's theorem, which says the derivative of a differentiable function has the intermediate value property. I've always been a bit dissatistied ...
1
vote
1answer
28 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 ...
5
votes
2answers
131 views

Prove that an annulus is not simply connected?

I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r < R$)$$ X = \{ ...
1
vote
2answers
53 views

Topology with one element

I am asked to prove that a topological space with one element is connected. My question: I am a little confused by what we mean by a topological space with only one element. Is this sort of ...
2
votes
0answers
70 views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
1
vote
2answers
42 views

The intersection of locally connected spaces is locally connected?

I am trying to prove that: Let $A$, $B$ subspaces X , if $A$ and $B$ are locally connected then $A \cap B$ is locally connected but I can't prove or disprove. I don't show my attempt because I have ...
2
votes
1answer
70 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, ...
3
votes
1answer
89 views

A metric space with a countable dense subset removed is totally disconnected?

I am wondering if it is true, and what the proof would go like, that given a metric space $X$ with a countable dense subset $D$, $\ X\setminus D$ is totally disconnected.
1
vote
2answers
36 views

Showing triangular region is connected

I'm trying to show that the region $R = \{(x,y) \in (a,b)\times(a,b) : x<y \}$ as a subset of $R^2$ is connected. I've tried to do this by showing that it is path-connected, (it clearly is) but how ...
0
votes
1answer
34 views

Union of two sets that are connected in the closure is connected.

Suppose $A$ and $B$ are connected subsets of a topological space $X$ such that $A \cap cl(B)$ is non-empty. I want to show that $A \cup B$ is connected. I can do this, but only if I assume that $X$ ...
1
vote
1answer
47 views

Connectedness and Continuity

Suppose $S^n\subset \mathbb{R}^n$ be the set of all points in $\mathbb{R}^n$ with unit Euclidean norm. That is $$S^n=\{x\in\mathbb{R}^n~:~\|x\|=1\}$$ If $f:S^{n-1}\rightarrow\mathbb{R},~n>2$, is ...
4
votes
0answers
48 views

What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?

I am working on an exercise problem about components and path components of $\mathbb{R}^{\omega}$. Specifically, Exercise about components and path components: 1. What are the components and ...
2
votes
1answer
66 views

Why is the “topologist's sine curve” not locally connected?

In Munkres's Topology, it is claimed that "The topologist's sine curve" is not locally connected without further explanation (See Example 3 of Section 25 "Components and Local Connectedness", 2nd ...
0
votes
3answers
32 views

Is a subspace connected?

$A= \{(x,y) \in \mathbb{R}^{2} : |y|>|x| \}\ \cup \ \{ (0,0)\}$ $B= \{(x, \frac{1}{x^{2}} ) \in \mathbb{R}^{2} : x>0 \} \ \cup \ (\mathbb{R} \times \{0\})$ Is it A or B connected? Why? I guess ...
1
vote
1answer
37 views

Connected subspaces and homeomorphism

$S_{1} = \{ (x,y)| y=\sin (\frac{1}{x}), 0<x \le 1 \}$ $S_{2} = \{ (x,y)| y=\sin (\frac{1}{x}), -1\le x <0 \}$ Let $S= S_{1} \cup S_{2} \cup \{(0,0)\}$ 1) Is S connected as a subspace ...
0
votes
1answer
49 views

On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
5
votes
1answer
34 views

Connectedness and compactedness of a sum 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 $A \cup B$ ...
4
votes
3answers
187 views

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 $\Bbb R^2 - X$ ...
0
votes
2answers
24 views

Prove proposition regarding connectedness

Let $(X,T)$ be a topological space, $A \subseteq B \subseteq \overline{A} \subseteq X$, $A$ is connected ($\overline{A}$ denotes the closure of $A$). Prove that $B$ is connected. I've just started my ...
0
votes
1answer
53 views

Is there a connected topology space that is disconnected by remove any point?

I wonder if there is a connected space $X$ such that $X\setminus\{x\}$ is disconnected (not necessarily totally disconnected) for every $x\in X$. update: This question comes from the case where I try ...
3
votes
2answers
123 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.
8
votes
2answers
112 views

Connectedness of the complement of a homeomorph of a ball

Let $n\geq 2$. My question is in two parts: If $B\subseteq\mathbb{R}^n$ is bounded and homeomorphic to the open unit ball in $\mathbb{R}^n$, is $\mathbb{R}^n\setminus B$ connected? Does the answer ...
1
vote
0answers
74 views

Connectedness in proximity spaces

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$. ...
2
votes
2answers
124 views

Locally path-connected implies that the components are open

If $X$ is a locally path-connected space, then its connected components are open. I am trying to prove this, but for some reason it doesn't seem right to me, knowing that components are always ...
2
votes
1answer
41 views

Question on separations of path-connected spaces.

The following questions showed up on a previous exam of mine, but I didn't know how to do it at the time. I'll post my work after the statement of the question, but was wondering if anyone could help ...
2
votes
0answers
75 views

Locally / Strongly connected product spaces

I have to give necessary and sufficient conditions s.t. $ X=\prod_i X_i$ is locally / strongly connected, where $(X_i,\tau_i)$ are non-empty top. spaces. First locally: Assume all $X_i$ are locally ...
3
votes
1answer
33 views

One component of border implies simply-connectedness

Is it right that any open connected bounded subset of $\mathbb{R}^2$ with one connectedness component (in usual or linear sense) of border is simply-connected? For me it seems to be kind of Jordan ...
2
votes
0answers
38 views

Examples of extremally disconnected spaces

I am trying to understand the notion of extremally disconnected space (in other words Stonean space), i.e. a space in which any open set has an open closure. Could you help me and give (reasonable) ...
4
votes
2answers
156 views

Connected and not path-connected

We have $\mathbb R^2$ (real plane) with the Euclidean topology. Define $X(n) = \{1/n\} \times [-n,n]$, all subspaces of $\mathbb R^2$. $$Y = \mathbb R^2 \setminus \bigcup_{n\ge1}X(n)$$ Prove that ...
5
votes
1answer
84 views

A topological property of shapes like $\bot$ in $\Bbb{R}^2$

Let $X$ be a shape like $\bot$ as a subspace of $\Bbb{R}^2$. Is it possible to construct a continuous function $f : X\times X \longrightarrow X$ such that for any $v,w \in X$, $f(v,w) = f(w,v) \in ...