Connected subsets one of which meets the closure of the other
Exercise 224 (Chapter 32, “Connectedness”):
Let and
be connected subsets of topological space
such that
intersects
. Prove that
is connected. Find an example to show that it is not enough to assume that
intersects
.
Let us assume that and
are open subsets of
such that
is empty and that
. To show that
is connected, we need to show that either
or
does not intersect
.
Since is connected, and
is empty and
, necessarily
is entirely in
or in
. Similarly,
is entirely in
or entirely in
.
Let us suppose that they are not both in or both in
; for instance, that
and
.
If , then
with
open in
. Since
does not intersect
,
cannot be in
. Hence
cannot intersect
. This contradicts our assumption.
Hence it is impossible that and
. Similarly, it is impossible that
and
. Thus,
and
are either both in
or both in
, which implies that either
or
does not intersect
.
If we only assume that and
intersect, it does not always follow that
is connected. For instance, let
be the topological plane and
and
two open disks that “almost touch”, that is such that the corresponding closed disks ‑ which are their closures ‑ intersect in one point. The union of
and
is then not connected.