Connectedness is not weakly hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) not satisfying a topological space metaproperty (i.e., weakly hereditary property of topological spaces).
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Contents
Statement
It is possible to have a nonempty topological space and a nonempty closed subset
of
such that:
-
is a connected space
-
is not a connected space under the subspace topology from
.
Related facts
- Connectedness is not hereditary includes more motivating discussion, the examples here are a subset of the examples on that page.
Proof
Example using finite topological spaces
For a counterexample, must have at least two points, because the unique one-point space is connected. Therefore,
must have at least three points. Below is one such example:
with the topology defined as follows: the open subsets are:
Or equivalently, the closed subsets are:
Clearly, is connected: the only nonempty closed subset containing
is all of
, and therefore
cannot be expressed as a union of two disjoint nonempty open subsets. In fact, framed more strongly,
is an irreducible space.
Consider to be the subset
of
with the subspace topology.
has a discrete topology, and in particular, is a union of disjoint closed subsets
and
. Therefore, it is not connected.
Basically, the point serves the role of connecting the space, and removing it disconnects the space.
Example using the real line and a finite subset
Consider the following example:
-
is the set of real numbers endowed with the usual Euclidean topology.
-
is a subset of size two.
is connected.
is discrete in the subspace topology. Explicitly, for instance,
is the intersection of
with the open subset
of
, hence is open in
, and similarly
and hence is open in
.