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Questions tagged [connectedness]

Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

A topological space is connected if it cannot be written as union of two disjoint non-empty open sets. Every topological space can be partitioned into connected components, which are connected subsets which are maximal with respect to inclusion.

Several related properties are studied in topology:

In graph theory, a connected graph is a graph such that there exists a path between any two vertices.

3082 questions
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Prove that $f$ is surjection.

$f:\mathbb{R}^n\to \mathbb{R}$ be a continuous map. $f$ not be a surjection if for all $c>0$, $f^{-1}(c)$ bounded. My idea: I guess I can use "if $A$ is a connected set, then $f(A)$ is connected." But I can't prove $f$ is a surjection.
J.Kan
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Is $\Bbb R^2 -\Bbb Q^2$ connected?

I need to prove whether or not $X=\Bbb R^2 -\Bbb Q^2$ is connected. The definition of connectedness I am using is a space X is connected if it is not the union of two disjoint nonempty open sets. This is a very new concept for me so I am not exactly…
bug123
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How to find a path in $SL(2)$?

$SL(2,\mathbb{R})$ is path-connected. Therefore, for all $A,B \in SL(2,\mathbb{R})$ there is a path $\varphi:[0,1]\rightarrow SL(2,\mathbb{R})$, connecting both matrices. I would like to know, how I can actually construct such a path for two given…
murphy
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How to prove $GL^+ (2, \mathbb{R})$ is connected

I am looking to prove that the set of $2\times 2$ matrices with positive determinant is connected. I understand that the set of invertible $2\times 2$ matrices is however disconnected since the determinant can not equal zero. I know That my aim is…
Mary
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Is $\mathbb{Q}^2 \cup \mathbb{I}^2 $ disconnected?

My intuition says that it is but I'm not entirely sure. I thought about using the projection map since it is continuous and surjective and also because I know that the rationals and irrationals are disconnected in the reals. However, there isn't a…
user328442
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Connected subset

I have the follwing question: Let $M$ be a connected metric space, $X \subset M$ is connected. Show that, if $A \subset M - X$ is both open and closed in $M - X$ then $A \cup X$ is connected. I don't know hou to proceed, because if $A \subset M - X$…
ThePoet
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Good Methods for Proving Connectedness

Is there a good standard way of proving connectedness? Disconnectedness seems a lot easier, as you just need to find the sets. For connectedness is there a similarly straightforward approach? I think I've heard path-connectedness is often…
user82004
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Proof of Interval is Connected

$ I \subset \Bbb R\; \text{is an interval} \Rightarrow I \subset \Bbb R\; \text{is connected}$ I am reviewing the lecture note about proving above statement. Proof is such as below: Suppose $I$ is a disconnected invterval. Then $I = u \cup v$, where…
Beverlie
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Locus of $xy=1$ is not path-connected

Artin, Algebra, Chapter 2, M6 It is intuitively easy but I am not sure if the writing my proof is rigorous enough. Let $a=(a_1,a_2,\cdots,a_k)$ and $b=(b_1,b_2,\cdots,b_k)$ be points in $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function…
velut luna
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Is $A = \{ (x,y): x^2 - y^2 = -1 \}$ path connected in $\mathbb R^2$?

Is $A = \{ (x,y): x^2 - y^2 = -1 \}$ path connected in $\mathbb R^2$ ? From its graph, I would conclude that it's not path connected. Take $(0,-1)$ and $(0,1)$ in A. Take any path $f: [0,1] \to A$ with $f(0) = (0,-1)$, $f(1) = (0,1)$. How can we…
user2015
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A very weird connected subset of $\mathbb R^2$

Is it possible to construct a connected subset of the plane with the property that removal of any single point makes it totally disconnected? Any answer is appreciated..Thanks!!
Aneesh M
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Which of the following sets are connected?

In the set of all $n×n$ matrices with real entries, considered as the space $R^{n^2}$ , which of the following sets are connected? (a) The set of all orthogonal matrices. (b) The set of all matrices with trace equal to unity. (c) The set of all…
Sriti Mallick
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General approach to connectedness proofs, with specific example

Connectedness proofs have given me trouble for far too long. Does anyone have good insights on how to go about proving a set is connected? I know that is a broad question, any insight at all would be helpful. I have provided a specific question…
Merkh
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Prove that $A$ is connected

Q. Prove that: If we remove 12 points from the Euclidean plane $\mathbb{R}^2$ to get set $A$, then $A$ is connected. I don't really understand how to proceed with the question. I thought of going via contradiction but I am stuck. Should I approach…
Sepia Glen
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What is the minimum number of configurations which connects all elements?

I am facing a problem which I can't seem to resolve. I'm no mathematician, so maybe I am missing something obvious. My problem is defined as follows: I have a list of 10 nodes (0,1,2,3,4,5,6,7,8,9), which all have to be connected to each other. A…
patrickvk
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