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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- Connectedness is not hereditary includes more motivating discussion, the examples here are a subset of the examples on that page.
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 .