Question

Verify that any infinite set in a discrete space is closed and bounded but not compact. [Hint: In such a space no sequence of distinct terms clusters.]

Short answer

An infinite set in a discrete space is closed, bounded, but not compact.

Step-by-step solution

Understanding a Discrete Space

A discrete space is a topological space where every subset is open. This means that each single point forms an open set by itself. In particular, every point is isolated, and there is no clustering of distinct terms.

Defining Closed and Bounded Sets

In a topological space, a set is closed if its complement is open. In a metric space, a set is bounded if there is a real number such that the distance between any two points in the set does not exceed this number. In a discrete space with usual metric (where the distance is 1 between distinct points and 0 with itself), any set is bounded since you can choose a bound larger than 1.

Proving Any Infinite Set is Closed

Let \( A \) be an infinite set in a discrete space. The complement of \( A \) is also open in the discrete space because every subset is open. Therefore, \( A \) is closed by definition, as its complement is open.

Demonstrating Boundedness

In a discrete space, any set is naturally bounded. For any infinite set \( A \), the distance between any two points in \( A \) is either 0 or 1 (according to the metric), so \( A \) is bounded with a bound of 1.

Discussing Compactness

A set is compact in a metric space if every open cover has a finite subcover. In a discrete space, an infinite set cannot be compact because there are no limit points. Any sequence composed of distinct elements has no convergent subsequences, as an infinite set of isolated points cannot be covered by a finite subcover. Each point must be individually covered.

Conclusion

In a discrete space, an infinite set is closed, bounded, but not compact, due to the nature of discrete topology and the lack of convergence of sequences.

Key concepts

Closed Sets

In topology, a closed set can be thought of as a set that contains all its limit points. However, in a discrete topology, this concept takes a simpler form. Since every point is isolated and every possible subset is open, any set can be considered closed by definition if its complement is also open.

This is easily satisfied in discrete topology because the nature of the space dictates that each subset of the space including the complement of the set is open. Thus, every subset is also closed. A practical takeaway is that closed sets in discrete topology are simply those whose complement subsets are part of the topological space's open sets.

• Closed sets contain all their limit points, but in discrete topology, they inherently possess this property because there are no limit points.
• Closed and open sets in discrete topology are interchangeable due to the openness of all subsets.

This characteristic greatly simplifies the concept of closed sets in such spaces.

Bounded Sets

Bounded sets, when discussed in the context of metric spaces, refer to sets where the distance between any two points does not exceed a certain bound. But what about in a discrete topology? Here, things become much simpler again!

In discrete spaces with the usual metric, the distance between distinct points is always 1, and there's no distance to cover when looking at individual isolated points of a set. This means that every set, regardless of size, is naturally bounded. The bound, in this case, is simply 1, as no two distinct points are further apart than this distance.

• Sets are bounded if a bound exists for the maximum distance between any two points.
• In discrete topology, this concept is trivially satisfied for all sets.

This allows discrete spaces to be uncomplicated when it comes to boundedness.

Compactness

Compactness is a more involved concept that combines both mathematical intuition and technical definition. A set is compact if, whenever it is covered by open sets (an open cover), there exists a finite selection of these open sets (a finite subcover) that also covers the set.

In discrete topology, however, compactness becomes less intuitive. Due to the innate property of discrete spaces having no accumulation points, infinite sets cannot be compact. Each element being isolated implies that any cover must address each point separately with an open set, preventing a finite cover from existing. Thus, compactness cannot be achieved.

• Compactness requires that from an open cover, a finite subcover can be chosen.
• Infinite sets in discrete topology cannot meet this requirement due to the isolation of points.

This makes compactness an elusive property for infinite sets in discrete spaces.

Topological Space

Topological spaces are foundational structures in topology, encompassing a set of principles that define how sets relate spatially. They provide a framework to conceptually handle concerns about limit points, continuity, and compactness.

In a discrete topology, however, the framework is simplified significantly. Every subset is considered open, and subsequently, the language of topological spaces becomes less complex but still deeply relevant. This simplicity arises because the discrete nature isolates every point, allowing endless flexibility in defining open and closed sets.

• Topological spaces are flexible frameworks for discussing set properties.
• The discrete topology simplifies this by uniformly declaring all subsets as open.

The unique properties of discrete topologies provide a distinctive platform for exploring basic topological concepts.