# Connectedness is not weakly hereditary

This article gives the statement, and possibly proof, of a topological space property (i.e., connected space)notsatisfying 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 .