# Connectedness is not 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., subspace-hereditary property of topological spaces).

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## Contents

## Statement

It is possible to have a nonempty topological space and a nonempty subset of such that:

- is a connected space
- is not a connected space under the subspace topology from .

## Related facts

## Proof

### Key proof idea

The key proof idea hinges on removing points from that serve to *connect* the space.

### Examples 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. We discuss two (related) examples of spaces of size three.

#### Space with one closed point and two open points

Consider:

with the topology defined as follows: the open subsets are:

Or equivalently, the closed subsets are:

Clearly, is connected: the only nonempty open subset containing is all of , and therefore cannot be expressed as a union of two disjoint nonempty open subsets.

Consider to be the subset of with the subspace topology. has a discrete topology, and in particular, is a union of disjoint open subsets and . Therefore, it is not connected.

Basically, the point serves the role of *connecting* the space, and removing it disconnects the space.

#### Space with one open point and two closed points

This is the same as the previous example, but with the roles of open and closed subsets interchanged. Explicitly:

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.

#### Note on the interchange of roles of open and closed

For a finite space, interchanging the roles of open and closed subsets defines a new topological space. The corresponding does not hold for infinite spaces in general.

### Examples from the real line

### Removing one point

Consider the following example:

- is the set of real numbers endowed with the usual Euclidean topology.
- is the subset of nonzero real numbers.

is connected. However, is a union of two nonempty disjoint open subsets: the negative real numbers and the positive real numbers. In fact, these are its two connected components. Therefore, is not connected.

#### Taking 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 .

#### Taking a dense subset

- is the set of real numbers endowed with the usual Euclidean topology.
- is the subset comprising rational numbers.