All Questions
Tagged with metric-spaces connectedness
358
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Connected Metric Spaces: Strategies
I am not really sure if my ideas in this topic are correct. Can anyone help me?
Finding the connected components of a metric space $X$.
Suppose there are two connected components $C_1, C_2$ of $X$. ...
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3
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82
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A countable metric space implies totally disconnectedness
I'm stuck on a problem in general topology. But first, let's establish some observations:
Let $(X, \tau)$ be a topological space. The connected component of $x \in X$, denoted by $C(x)$, is the union ...
- 135
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54
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Accessibility Components does *not* coincide with Path Connected and Connected Components for Length Spaces
I am working through "A Course in Metric Geometry" and I do not believe this exercise to be true:
Exercise $\boldsymbol{2.1.3}$
$3)$ Verify that accessibility components coincide with both ...
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Simply connected set
An open connected set $S$ in $\mathbb{R}^2$ is said to be simply connected if its complement relative to the whole plane is connected. This definition is mentioned as an equivalent definition in the ...
- 431
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Confusion about connectedness
A metric space is said to be "connected" it there is no subset that is both closed and open.
Sets are open if "any sufficiently tiny ball around a point is fully contained" and &...
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Help proving that this metric space is not path connected
Consider the metric space embedded in $S^1$ with the intrinsic metric(the distance between two points is the length of the shortest arc connecting them):
$\hspace{3cm}$
Notice there are $3$ 'gaps' in ...
- 3,044
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66
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Is an open connected subspace of $\mathbb{R^2}$ locally path-connected?
I want to proof the following statement :
Let X be an open connected subspace of $\mathbb{R^2}$. Show that X is also path connected. The standard way to prove this problem from what I at least saw was ...
- 405
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Connected components and limited sets
$X \in \mathbb{R}^{n}$ is a limited subset. Prove that $\mathbb{R}^{n} - X$ has exactly one ilimited connected component, for $n > 1$.
I tried to show that if $\mathbb{R}^{n} - X$ has more than one ...
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Connected Sets and Intermediate Value Property
$f: M \to \mathbb{R}$ is a continuous function. If $c \in \mathbb{R}$ is a number strictly comprehended between the maximum and the minimum of $f$ in M, then $M - f^{-1}(c)$ is not connected.
I think ...
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Continuous extension of a continuous function $f:A \subseteq [0,1] \rightarrow X$ where $A$ is closed and $X$ is a peano continum
A peano continuum is a connected, compact and locally connected metric space. This kind of spaces is arc connected and as is stated in the following is uniformly locally arc connected.
The problem is ...
- 802
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Metric space that can be written as the finite union of connected subsets but isn't locally connected
I'm looking for an example of a metric space $X$ such that for every $\epsilon > 0$ there exist connected subsets $A_1, \dots A_n$ for some $n \in \mathbb{N}$ such that $X = \cup_{i = 1}^nA_i$ and ...
- 802
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66
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Is this subspace of $\mathbb{R}^2$ connected? arcwise connected?
Let
$$
A \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, 0 < x \leq 1, \ y = \sin \frac1x \right\}
$$
and
$$
B \colon= \left\{ (x, y) \in \mathbb{R}^2 \colon \, y=0, \ -1 \leq x \leq 0 \right\},
$...
- 26.9k
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Does locally compact, locally connected, connected metrizable space admit a metric with connected balls?
If $X$ is a locally compact, locally connected, connected metrizable space, does that imply that there must be a metric $d$ on $X$ such that $B(x, r) = \{y\in X : d(x, y) < r\}$ is connected for ...
- 10.5k
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Let A be a countable set of $\mathbb{R}^n , n>1$. We claim that $\mathbb{R}^n-A$ is connected.
The proof goes as follows... "Given $x,y \in \mathbb{R}^n-A,$ Choose a point $z \in [x,y]$ other than $x$ and $y$, where $[x,y]=\{a \in \mathbb{R}^n :a=(1-t)x+ty,0\leq t\leq1\}$, Now choose $w \...
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$\Bbb R\setminus \Bbb Q$ the set of irrationals is disconnected [duplicate]
I'm trying to find two non-empty disjoint open set $A, B$ such that $A \cup B = \Bbb R \setminus \Bbb Q$.
But can't find
