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Edit: Consider the notion of "simply connected," and drop the "connected". Then we obtain the following concept.

Call a subset $A$ of a topological space simple if for all $x,y \in A$ such that there exists a path between $x$ and $y$ lying completely in $A$, it holds that any two such paths are equivalent up to continuous deformation.

Is it always the case that the arbitrary intersection of simple subsets of $\mathbb{R}^2$ is itself simple? Note that this fails if $\mathbb{R}^2$ is replaced by the sphere.

If so, define that the simplification of any $A \subseteq \mathbb{R}^2$ is the intersection of all simple supersets of $A$. If $A$ is connected, is its simplification necessarily connected?

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"Simple" is commonly known as "path connected". – Asaf Karagila Feb 23 '13 at 21:44
I think you've misread the question, I've dropped connectedness altogether. – goblin Feb 23 '13 at 21:45
Simple if there is a continuous function $\gamma\colon[0,1]\to A$ such that $\gamma(0)=x$ and $\gamma(1)=y$? – Asaf Karagila Feb 23 '13 at 21:47
So each path-component of $A$ is simply connected? – Ben Millwood Feb 23 '13 at 21:51
@BenMillwood: No, it is not. As $A$ take the unit circle $S^1$ and consider the functions $\exp(t2\pi i)$ and $\exp(2t2\pi i)$ -- they're not homotopic. – Damian Sobota Feb 23 '13 at 23:06
up vote 1 down vote accepted

The result becomes easy by using a nice (not fully trivial) result:

A set $A\subseteq \mathbb R^2$ is simple if and only if its complement in $\overline{\mathbb R^2}:=\mathbb R^2\cup\{\infty\}$ is connected.

Now if $A_i$ is simple for each $i\in I$, then $$ \overline{\mathbb R^2}\setminus \bigcap _{i\in I} A_i = \bigcup_{i\in I}\left(\overline{\mathbb R^2}\setminus A_i\right)$$ is the union of connected subsets that all have the point $\infty$ in common and hence is connected.

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Is there a reference where I can see the proof of the (not fully trivial) result? And if so, for which spaces is it proved? Presumably $\mathbb{R}^n$, but perhaps others, too? – goblin Feb 25 '13 at 1:55

An arbitrary intersection of simple sets is simple: if $\gamma, \gamma'$ are two coterminal paths in the intersection, then they belong to each of the original sets, and so the interior of the set bounded by these paths belongs to each original set; this holds (and is well-defined to begin with) because we're dealing with $\mathbb{R}^2$. Therefore the interior belongs to the intersection, and hence the two paths can be continuously deformed to one another within the intersection.

The fact that the simplification of a connected $A$ is connected is trivial. Suppose $S$ is its simplification. Let $S'$ be the connected component of $S$ which contains $A$; there must be a single one because $A$ is connected. Then $S'$ is a union of path-components of $S$ (since path-connected implies connected), and thus $S'$ must be simple as well, so $S \subseteq S'$. Well then $S = S'$, so the simplification of $A$ is connected.

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