3
votes
2answers
41 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
2
votes
3answers
70 views

Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
1
vote
1answer
19 views

The intersection of a connected subspace with the boundary of another subset

Can someone please verify my proof or offer suggestions for improvement? Definition/Notation: The boundary of $A$, denoted by $\operatorname{Bd}(A)$, equals $\overline{A} \cap \overline{X-A}$. ...
1
vote
2answers
35 views

Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
0
votes
1answer
44 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
1
vote
1answer
36 views

Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
1
vote
1answer
41 views

Connectedness of both $Y \cup A$ and $Y \cup B$ where $A, B$ is a separation of $X -Y$

Let $Y\subset X$ be such that both $X$ and $Y$ are connected. Show that if $A$ and $B$ is a separation of $X-Y$, then $Y\cup A$ and $Y\cup B$ are connected. I found a proof for this problem in this ...
1
vote
1answer
25 views

Path Connectedness argument for $SO(n, \mathbb{R})$

I am trying to prove path connectedness of $SO(n, \mathbb{R})$. I have seen several different proofs for the same. But I had a thought and wanted to know whether it would help in any way. I took two ...
3
votes
1answer
40 views

Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?

Question (Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must ...
1
vote
2answers
40 views

Connected Sets in Topology

Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected. (Cl(E) is closure of E) Question: ...
0
votes
1answer
70 views

Is the sphere $S^n$ always arcwise connected?

I have a small question about the connectedness of the sphere; Is the sphere $S^n$ always arcwise connected ? Thank you.
2
votes
1answer
34 views

Simple question on connectedness in a subspace [duplicate]

For some reason I am having some trouble on this basic point set topology question: Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, ...
6
votes
1answer
596 views

Proof: X is connected

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
2
votes
1answer
53 views

Connectedness and some properties [closed]

Let $S \subset \mathbb R^2 $ be defined by $$ S = \left\{ \left(m + \frac{1}{2^{|p|}}, n + \frac{1}{2^{|q|}}\right) : m,n,p,q \in \mathbb Z \right\}.$$ Is $S$ discrete? Is $\mathbb R^2\backslash S$ ...
-2
votes
1answer
62 views

Prove that intersection of connected spaces is connceted.

Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected. Using the definition of connected space is that the ...
5
votes
1answer
48 views

Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that ...
5
votes
1answer
57 views

A maximal subset of $S^2$ with respect to a connectedness property

Let the set $A$ be a circle with a chord on the sphere $S^2$. Obviously $A$ has the following property: P: $\quad$ Any two points $a$ and $b$ of $A$ can be connected by a path that ...
0
votes
0answers
32 views

Disconnecting using totally disconnected sets [duplicate]

Let $X$ be $[0,1]^2$ and $S\subset X$ a totally disconnected subset. Is it true that $S^c$ is always connected? If it is false, what can we say when $X=[0,1]^n$?
2
votes
1answer
58 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 ...
3
votes
1answer
42 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
1
vote
0answers
71 views

Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
2
votes
1answer
37 views

Product of Connected Spaces (2)

If $Y$ and $Z$ are connected, is $Y \times Z$ path connected? I cannot find a counter example. Some help please.
4
votes
0answers
57 views

When does connectedness imply path-connectedness

In a locally path-connected space connectedness and path connectedness are equivalent. What is the minimal condition we would impose on a topological space to get the same result?
4
votes
1answer
50 views

$A\cup B$ is connected when $A$ is connected in $X$ and $B$ clopen in $X-A$

Let $A$ be connected subset of a connected space $X$, and $B\subset X-A$ be an open and closed set in the topology of the subspace $X-A$ of the space $X$. Prove that $A\cup B$ is connected. I think ...
1
vote
0answers
43 views

Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
1
vote
1answer
69 views

Countable subsets are disconnected

I am trying to show the following: every countable subset of $\Bbb R$ with at least two points is disconnected. My attempt: let $D$ be such subset. Then take $a \in D$ and define $A=\{ a\}$ and $B = ...
2
votes
1answer
66 views

is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of ...
0
votes
1answer
64 views

Proof: The quotient space is totally disconnected

I want to prove the following: Let $X$ be a topological space. Remark: $x \sim y \iff$ There is a connected component which contains $x$ and $y$. And now I want to show that the quotient space ...
3
votes
1answer
33 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
3
votes
2answers
111 views

Arcwise connected but not connected?

In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces: With a few pathological exceptions, arcwise connectedness is practically ...
9
votes
1answer
108 views

Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with ...
0
votes
1answer
18 views

Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is M path connected?

The Assignment: Let $K_1(0) := \{x \in \mathbb{R}^2: \|x\|_2 < 1\}$ and $S := K_1(0) \setminus \mathbb{Q}^2$. Is S path connected? Explain your answer. I don't think S is path-connected since ...
0
votes
1answer
124 views

Hyperspace and connectedness

I'm looking for any theorems and proofs for connectedness for hyperspaces exp(X). I would like to take a look for especially this theorem: $$ X \textit{ is connected } \leftrightarrow exp(X) ...
1
vote
1answer
68 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
4
votes
0answers
54 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
1
vote
3answers
72 views

If n > 1 and $B \subset \mathbb R^n$ countable. Then $\mathbb R^n - B$ is connected ( James Dugundji)

We can assume that $0 \in B$ , otherwise we move the origin.. We show that the origin and $ x \in \mathbb R^n - B$ are contained in a connected set lying in $\mathbb R^n - B$. Draw ...
3
votes
2answers
194 views

Countable basis but uncountably many connected components

Looking for some guidance on two topology questions: (a) Show that a locally connected space with a countable basis, has at most countably many connected components. (b) Give an example when X has ...
0
votes
1answer
27 views

Show that set is path connected?

How do I show that the set $A = \{(x,y) \in R^2: x \geq 0, y \geq 0\} \cup \{(x,y) \in R^2: x \leq 0, y \leq 0\}$ is path connected. I know that I need to construct a continuous function $f:[0,1] ...
0
votes
1answer
35 views

Finding the connected components of topological spaces

Find the connected components of the following sets: $(a) \; A=\{(x,y):y=\sin(1/x), x\in\mathbb{R}^+\},(b)\;A\cup\{(x,y):x=0,y\in[-1,1]\},(c)\;$The Cantor Set,$(d)\;\mathbb{N}$ with the cofinite ...
11
votes
1answer
71 views

Deleting $n$ points from a connected space

Let $X$ be a space such that for any subset $S \subset X$ with finite cardinality $n$, the subspace $X \setminus S$ has exactly $n+1$ connected components, each of which is homeomorphic to $X$. Is ...
2
votes
3answers
203 views

Möbius band with its middle part removed is still connected

Let $I\times I/(0,t){\sim}(1,1-t)$ be the Möbius band and let $S=\{(x,y): (x,y)\in M, 1/4<y<3/4\}$ be its middle part. How can I show that $M-S$ is connected? I tried to write a continuous ...
0
votes
1answer
25 views

Can a connected subpsace be in disjoint open sets?

Say $X$ is a non-connected topological space, i.e. $X= U\cup V$, and $U,V$ are disjoint (non-empty) open sets. Then suppose $C$ is a connected subspace of $X$, with the standard subspace topology. Can ...
1
vote
1answer
136 views

Connectedness problem on the 2-sphere

Let $K \subset L \subset S^2$, where $S^2$ is the 2-sphere and $K$ and $L$ are compact subsets with empty interior and $L$ is connected (I don't think all of those are relevant hypotheses though, but ...
0
votes
1answer
33 views

Examples of continuous integer-valued functions on totally disconnected spaces

I wanted to see examples of continuous integer-valued functions $f:X\to \mathbb{R}$ on a totally disconnected space $X.$ I have only some abstract examples in mind.
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 ...
1
vote
1answer
35 views

Dugundji problem about quasicomponents.

Problem 5 part d), of chapter V, section 3 (p. 118): d. In $E^2$ let $L_1$ be the line $x=1$ and $L_2$ the line $x=-1$. For each $n\in\Bbb Z^+$ let $R_n$ be the rectangle $\{(x,y):|x|\le ...
2
votes
1answer
32 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 ...
2
votes
2answers
90 views

Is a continuous image of $S^1$ to Hausdorff space locally connected?

Is a continuous image of $S^1$ to Hausdorff space locally connected? How do you prove this?
4
votes
1answer
82 views

Prove that local connectedness is preserved by continuous closed functions

I thought about applying the proof for open functions, namely: Proposition: If $f$ is a continuous open function from a locally connected space $X, \tau$ onto a space $T, \tau'$, then $Y$ is locally ...
2
votes
1answer
25 views

Inner product space is connected

How does one show any inner product space is connected? Shall I start with assuming that it is not connected, and arrive at a contradiction? So let $X$ be an inner product space, and there exist open, ...