Regions are Connected
It is obvious that regions are connected, but proving it took a while, because it’s very easy to use arguments that are ‘obvious’. This is an easy problem, but I really like my proof, so here it is:
Let 
be a region, and assume 
for non-empty sets 
and 
such that 
. Let 
and 
.
Since 
is convex (clear by the triangle inequality), all points of the form 
where ![t \in [0,1]](/cz/latexrender/pictures/97dea654b5094f652b139f64f97a5acc.png "t \in [0,1]")
lie in . Let ![T = \{ t \in [0,1]\; :\; y(t) \in U\}](/cz/latexrender/pictures/8e86de10f8714981f7d484afa1ffd297.png "T = \{ t \in [0,1]\; :\; y(t) \in U\}")
; clearly 
and 
. Likewise, if ![S = \{ t \in [0,1] \; : \; \forall s \in T:\; t>s \}](/cz/latexrender/pictures/f6e113dd2a0381ae287ae78fde19c1a4.png "S = \{ t \in [0,1] \; : \; \forall s \in T:\; t>s \}")
, then 
.
Either 
contains its least upper bound, or, exclusively, 
contains its greatest lower bound. In the first case, contains a sequence converging to 
; in the second, contains a sequence converging to 
. Therefore, either there is a sequence of points in converging to a point in or vice versa. This shows 
or 
, therefore is connected.
Possibly relevant posts:
- Pointwise convergence implies weak convergence (9/26/2008)
- Every finite spanning set is a frame (5/26/2006)
- Not a frame, but Bessel (5/26/2006)