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Definition for path connected: A topological space $X$ is path connected if for every $x, y \in X$ there is a path in $X$ from $x$ to $y$. A subset $A$ of a topological space $X$ is path connected in $X$ if $A$ is path connected in the subspace topology that $A$ inherits from $X$.

Now, Consider $\mathbb{R}^2$ with the standard topology and define two sets $A$ and $B$ on $\mathbb{R}^2$ as:

$A \subset \mathbb{R}^2$ given as $A = \{(x,y) \ s.t.\ –1 < x < 1\ and\ –3 < y < x\}$
$B \subset \mathbb{R}^2$ given as $B = \{(x,y) \ s.t.\ –1 < x < 1\ and\ x < y < 5\}$.
Both A and B are path connected.

I've been reading some other Questions like this one, and here's a few examples of sets that is also path connected: $A \cup cl(B)$, $cl(A) \cup cl(B)$, and $A \cup B$.

Are these correct? Thanks.

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P Haggerty
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  • Note, per comments on the Accepted Answer that $A\cup B$ is not path connected (not even connected). – hardmath Dec 24 '16 at 12:09
  • $A , B, A\cup \overline B,$ and $\overline A \cup \overline B=\overline {A\cup B}$ are convex sets and are therefore path-connected. $A\cup B$ is not connected because $A, B$ are open,not empty, and disjoint in the space $A\cup B$ and their union is the whole space ($A\cup B$). – DanielWainfleet Dec 28 '16 at 09:10

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Yes, these are fine because $A$ and $B$ are well behaved subspaces. But do not get too comfortable with assuming that the closure of a path connected space is again path connected.

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Andres Mejia
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