Understanding Connectedness in Topology: Why Open Subsets Can be Disconnected

In the study of topology, the concept of connectedness is fundamental. However, a common misconception is that an open subset of a connected topological space must also be connected. This article will explore this notion and provide concrete examples to clarify the misunderstanding. We will specifically look at the subset ( (0,1) cup (1,2) ) of the real numbers with the usual topology to illustrate that an open subset can indeed be disconnected.

Introduction to Connectedness

A topological space is considered connected if it cannot be expressed as the union of two disjoint nonempty open sets. In other words, a space is connected if it is not possible to separate it into two parts such that each part is both open and nonempty.

Connectedness in General

Let's first consider the real line (mathbb{R}) with the usual topology. This space is connected because any attempt to separate it into two nonempty open sets fails. For instance, if (mathbb{R}) were expressed as a union of two disjoint nonempty open sets, one of these sets would have to contain either (infty) or (-infty), which is impossible in the context of (mathbb{R}) with its standard topology.

The Counterexample: ( (0,1) cup (1,2) )

While the real line (mathbb{R}) is connected, an open subset within it can indeed be disconnected. This is precisely what the subset ( (0,1) cup (1,2) ) demonstrates. To see why, let's break it down:

Understanding the Subset ( (0,1) cup (1,2) )

The subset ( (0,1) cup (1,2) ) of the real numbers with the usual topology is given by the union of two disjoint intervals: ((0,1)) and ((1,2)).

Disconnected Nature of the Subset

Let's verify why ( (0,1) cup (1,2) ) is not connected. As a set, it is the union of the two open sets (intervals) ((0,1)) and ((1,2)). These intervals are disjoint, nonempty, and both open in the subspace topology inherited from (mathbb{R}). Therefore, ( (0,1) cup (1,2) (0,1) cup (1,2) ) can be expressed as the union of two disjoint nonempty open sets, thus making it disconnected.

Further Exploration

This raises an important question: if a space can be connected and have a disconnected open subset, what conditions must an open subset satisfy to be connected? The answer is that an open subset of a connected space does not need to be connected. The key is that connectedness is a property of the entire space, not its subsets.

Conclusion

Understanding the distinction between the connectedness of a space and the connectedness of its open subsets is crucial in topology. The subset ( (0,1) cup (1,2) ) in the real line with the usual topology serves as a clear counterexample to the notion that an open subset must be connected. This insight underscores the importance of careful consideration of topological properties and the need to distinguish between different levels of structure within a space.