# Questions tagged [path-connected]

Use this tag for question on path-connected spaces and related notions. These include locally path-connected spaces, arcwise connected spaces and so on. For the more general notion, use the (connectedness) tag.

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### Can we describe the connected components in the graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$?

The graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$ is, perhaps unsurprisingly, pretty wild. Here is a Desmos version of it; a few screenshots are below, showing different ...
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1 vote
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### Is there a nontrivial LOTS that is connected and totally path disconnected?

By nontrivial LOTS I mean a linearly ordered space that contains more then one point. Being totally path disconnected means that every path in the space is constant. A connected linearly ordered ...
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### Proof verification: Demonstrating that the unit circle is path-connected

I'm trying to show that the unit circle is path-connected. I have a sketch of a proof, and was hoping someone could tell me if this is on the right track. Given points $(a,b), (m,n)$ on the circle, I ...
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### Artin, Chapter 2, Misc.6

I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, Algebra. Below is the statement of the problem. Let $a = (a_1, \ldots, a_k)$ and $b = (b_1, \ldots, b_k)$ be points in $k$-...
• 297
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### Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected [duplicate]

Show that $\{(x,y): 0 < x \leq 1; y = \sin{1/x}\} \cup \{z: x = 0; -1 \leq y \leq 1\}$ is not path connected. The book gives this set as an example of a closed, connected but not path connected set....
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### Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
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### Is X connected by paths?

I have the following space $X=[0,1) \cup \{y,z\}$ with $y \neq z$ and $y,z \notin [0,1)$. Where the basic open for point in $[0,1)$ are taken as the basics to the topology that R inherits to the space....
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### Understanding the definition of path connected subspace

If $X$ is a topological space and $Y$ is a subspace of $X$, then what does it mean when we say $Y$ is a path connected subspace of $X$? Does it mean that any two points $x$ and $y$ in $Y$ ...
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### Locally connected for sub space topology of $\Bbb R^2$

Consider the following subspace $X:=\Bbb R^2\setminus(\Bbb Q\times \Bbb Q)$ of $\Bbb R^2$, where $\Bbb R^2$ with the usual topology. I would like to check this space in the terms of various kinds of ...
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