Question

If \(\left\\{X_{\alpha}\right\\}_{\alpha} \in A\) is a family of topological spaces of which infinitely many are noncompact, then every closed compact subset of \(\prod_{\alpha \in A} X_{\alpha}\) is nowhere dense.

Short answer

Closed compact subsets in the product space are nowhere dense.

Step-by-step solution

Identify Key Concepts

First, recognize that the problem involves understanding compact spaces and their relationship in product spaces. The statement talks about closed compact subsets being nowhere dense in the product space when infinitely many of the spaces are noncompact.

Review Definitions

Recall that a set is nowhere dense if its closure has empty interior. Hence, we need to show that any closed compact subset of the product space has an empty interior.

Understand the Product Topology

In the product topology, a basis is formed by the products of open sets in each component space. Since infinitely many of the spaces are noncompact, they cannot include compact set components covering all spaces.

Analyze the Intersection of Open Sets

Consider an open set in the product space. If a closed compact set has a nonempty interior, it would intersect a basis element of open sets in the product topology. However, noncompactness in infinitely many spaces implies that no finite subcover can exist for compact coverage.

Apply Compactness and Noncompactness

Using compactness, we argue that if a closed set were dense, it should fit into finite sub-intersections of open sets, but this conflicts with the noncompactness of infinitely many spaces. Thus confirming the closed set cannot be of full interior or even anywhere dense.

Conclusion

Thus, since the set cannot be anywhere dense due to the infinite noncompact spaces within the product topology, every closed compact subset of the product space is nowhere dense.

Key concepts

compact spaces

Compact spaces are foundational in topology, known for their neat and manageable properties. A compact space can be thought of as a generalization of closed and bounded sets in Euclidean space. To understand compactness:

• A space is compact if every open cover has a finite subcover.
• This means you can cover the entire space with open sets, but only need finitely many of them.

Compactness is crucial as it ensures certain desirable properties, such as every sequence in a compact space having a convergent subsequence.
In the context of product topology, compact spaces take on a new characteristic. It's important to note that in a product of infinitely many spaces, compactness isn't guaranteed unless all individual spaces are compact.
In the given exercise, we studied product spaces with infinitely many noncompact components. Here, the lack of compactness in these components influences the behavior of subsets, making it impossible for any closed, compact set to become 'large' in terms of density.

nowhere dense sets

Nowhere dense sets are quite interesting within topology because they essentially lack 'thick' presence in any space.
To define a nowhere dense set:

• A set is nowhere dense if its closure has an empty interior.
• This means the set is not only small in some sense, but even its closure cannot occupy a 'fat' neighborhood.

In simple terms, a nowhere dense set can be very spread out and has holes sufficiently large that it cannot contain any open set fully.
In our exercise relating to product topology, this concept plays a critical role. When dealing with a product of spaces where infinitely many are noncompact, closed compact sets in such products are nowhere dense.
This result stems from the inability of these compact sets to cover open regions due to the infinite noncompact dimensions. It's as if they fail to engage deeply with the topological space, always missing some part of the interior.

topological spaces

Topological spaces form the backbone of modern topology. They open a playground for defining and understanding various continuity concepts, connectedness, and limits.
Let's take a stab at what makes a space 'topological':

• It consists of a set equipped with a topology, a collection of "open" subsets that satisfy certain conditions.
• The whole set and the empty set must be part of this topology.
• The union of any collection of open sets and the finite intersection of open sets also belongs to it.

This structure allows mathematicians to generalize the notion of closeness, continuity, and boundary.
In product topology, which is crucial for understanding more complex spaces formed by combining simpler ones, the topology is defined as the coarsest one where projections are continuous. If any one component is noncompact, certain desirable properties, like compactness, disappear.
The product topology encapsulates the complexity of merging topological spaces, wherein every subset's behavior might change significantly compared to its behavior in the individual spaces. This intricacy is the key to understanding the constraints met in our problem regarding nowhere dense sets and compact spaces.