# 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.

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### Connected set which is no-where path connected

Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path. My solution to the puzzle was to use an order topology on a ...
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### Relation between connectedness and Dedekind Completeness

In my calc class we constructed the real numbers using the following 5 axioms: The set is nonempty. The set has an ordering. The set has no first/last point. The set is connected. The set contains a ...
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### Boundary of an open bounded connected subset of $\mathbb{R}^2$ must contain the image of a path?

How does one prove that the boundary of a bounded open connected subset of $\mathbb{R}^2$ must contain the image of a path? I require a non-constant path. But it doesn't have to be injective. This ...
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### What ensures the existences of such sequence

Let $(X,d)$ be a metric space and $A \subset X$ a connected subset. Let $B \subset X$ s.t. $A\subset B \subset \overline A$. Then $B$ is connected. I have a proof in my textbook that I understand ...
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### Can quasi-concave and monotone functions level curves that are not path-connected?

For $X = \mathbb{R}^2$, does there exist a quasi-concave and monotone function $f : X \to \mathbb{R}$ that has a level curve which is not path-connected? Secondly, will every level curve necessarily ...
1 vote
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### Path connectedness equivalent to connected in euclidean topology for quasiprojective vareities?

Let $V$ be a quasiprojective real variety. My intuition tells me that this type of space with the Euclidean topology has the property that path-connectedness and connectedness are equivalent. Is this ...
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### Connected and Locally compact

Im having some trouble with the following: Let $X=(0,1)$ and $T=\{(0,1-\frac{1}{n}|n \in \mathbb{N},n\geq 2\} \cup \{X,Ø\}$ I want to prove that (X,T) is: Connected The definition of being connected: ...
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### Is the space of $n\times n$ real symmetric matrices with strictly positive determinant connected within the vector space of $n\times n$ real matrices?

I want to make clear that I am aware of the connectedness in the case of general real matrices. But here I ask about the subspace of symmetric ones. If it is not the case, which are the connected ...
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### If $X$ has finitely many connected components then any connected component is open. Which subset in $X$ are open and closed?

I am currently reading Topology and for guidance I'm following my college lectures and the book by Munkres. I'm still in the beginning so I'm having a difficulty to understand the following : If $X$ ...
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### Homological vs. homotopical connectivity

There are two closely-related measures of connectivity of a topological space: Homotopical connectivity is the largest $n$ such that all homotopy groups up to and including the $n$-th one are trivial;...
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### Find a continuous map to show a set in $R^2$ is connected

I want to show that the set $S = \{(x,y) \in \mathbb{R}^2: 0 \leq x \leq y^2, 1 \leq y \leq 2 \}$ is connected My first take is trying to show that $S$ is the image of a continuous function $f$ ...
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### Exercise 6, Section 23 of Munkres’ Topology

Let $A\subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$ , then $C$ intersects $\operatorname{Bd}A$. My attempt: Approach(1): Assume towards contradiction, ...
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Let we have $S$, a connected surface. Let's suppose that the curvature of all its geodesic curves is constant, $K=1$. I have to prove that $S \subseteq \mathbb{S}^2$. $\mathbb{S}^2=\{(x,y,z)\in \... 0 votes 0 answers 19 views ### Prove uniqueness via a connectedness argument I am reading an article about uniqueness of some evolution PDE and at some point, the author says that to show uniqueness on an interval$[0,T]$, by a "classical connectedness argument" (his ... 3 votes 1 answer 48 views ### Disconnected image of a derivative I would like to know if there exists a differentiable vector-valued function$f:[a,b]\rightarrow \Bbb R^2$such that the image$f'([a,b])$of its derivative is disconnected. My Attempt First of all I ... 2 votes 1 answer 93 views ### Counterintuitive topological result concerning connectedness Problem If we consider$A,B,C,D$to be the corners of the square$[0,1]^2$where$A$and$C$are diagonally opposite (and so are$B$and$D$). Then we can define$X$to be a connected subset of the ... 0 votes 1 answer 41 views ### Counterexample of a disconnected topological space with open components but not locally connected Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open. What obviously follows from the theorem is that the components of a locally connected ... 4 votes 2 answers 170 views ### Intuitively, what is the difference between a "simply connected" set and a "locally connected" set? I was looking into the Mandelbrot Set and saw a note that said it has been proven that the Mandelbrot Set is "simply connected" but it is still an open question of whether or not it is "... 1 vote 1 answer 56 views ### Prove or disprove that a Jordan curve exists that contains two points. Question (not homework): Let$\Omega$be open and connected. Prove or disprove that if$z_1, z_2\in\Omega$are distinct, then there exists a Jordan curve$\gamma\subseteq\Omega$such that$z_1, z_2\...
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Let $X$ and $Y$ be two topological spaces. We denote with $CX$ and $CY$ the sets of connected components and $QX$ and $QY$ the sets of quasicomponents for the corresponding spaces. It is well known ...
Prove that if $f: X \rightarrow Y$ is a continuous function of metric spaces and $X$ is connected, then $f(X)$ is also connected. I have tried to prove this by the contrapositive assuming \$f(X) = A \...