Tagged Questions

41 views

What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?

Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected? This is not homework but is in response to working several problems where countable or ...
72 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
20 views

Two intersecting disks with subspace topology in $\mathbb {R}^2$

Take two closed disks as subsets of $\Bbb R^2$ such that they intersect at exactly one point. Let $\Bbb R^2$ have the standard euclidean topology $\mathcal J_E$ and give the above set the subspace ...
168 views

Is bijection mapping connected sets to connected homeomorphism?

If $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a bijection, mapping connected sets to connected, is $f$ necessarily a homeomorphism? The converse is true, a well known property of homeomorphisms. I ...
35 views

if $A\subseteq \mathbb R^n$ is connected then is $A´$ (derived set) connected?

are the following statements true? 1) if $A\subseteq \mathbb R^n$ is connected then is $A´$ connected? 2)if $A´$ is connected then is $A$ connected? I can´t find any counterexamples. Can you help ...
10 views

Examples of monotone mappings?

I am looking for some interesting (non-trivial) examples of functions between normal spaces which are perfect and monotone, i.e., functions which are surjective and closed preimages of singletons ...
27 views

Help with continuum theory

A continuum is a compact connected Hausdorff space (sometimes metric is included in the definition). I have yet to find any references that help me understand composants and components of a ...
37 views

70 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 ...
67 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 ...
38 views

71 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?
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 $$\text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i)$$ ...
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 ...