Question

Let \(B \subseteq A \subseteq(S, \rho)\). Prove that \(B\) is connected in \(S\) iff it is connected in \((A, \rho)\).

Short answer

B is connected in set S if and only if it is connected in subset A.

Step-by-step solution

Understanding the Concept of Connectedness

A subset is connected if it cannot be split into two nonempty, disjoint open subsets. Our task is to prove that if a subset \( B \) is connected in a larger set \( S \), it is also connected in any subset \( A \) containing it, and vice-versa.

Proving "If" Direction (Connected in \(S\) implies Connected in \(A\))

Assume \( B \) is connected in \( S \). If \( B \) could be split into two disjoint open sets in \( A \), say \( U \) and \( V \), then \( U \) and \( V \) would also be open in \( S \) when restricted to \( B \), contradicting the connectedness of \( B \) in \( S \). Therefore, no such split exists, implying \( B \) remains connected in \( A \).

Proving "Only If" Direction (Connected in \(A\) implies Connected in \(S\))

Conversely, assume \( B \) is connected in \( A \). Open sets \( U, V \) in \( S \) intersecting \( B \) would either have one of the sets empty or cover all of \( B \). Since \( A \) is part of \( S \) and \( B \) is connected in \( A \), \( B \) cannot be split into disjoint open sets in \( S \), hence \( B \) is connected in \( S \).

Concluding the Proof

Both the above directions show that \( B \) is connected in \( S \) if and only if it is connected in \( A \). Thus, this completes the proof of the statement.

Key concepts

Subset

In topology, the term "subset" refers to a set where each of its elements also belongs to another set. If we have a set \( S \) and another set \( B \), and every element found in \( B \) is also contained within \( S \), then we say that \( B \) is a subset of \( S \). This is denoted visually as \( B \subseteq S \).
Subsets play a crucial role in understanding more complex structures such as topology spaces. When dealing with properties like connectedness, understanding how subsets relate to their parent sets helps us in investing how properties within a larger set apply to the smaller ones within it. In our proof, \( B \) being a subset of both \( A \) and \( S \) forms the foundation of our arguments about connectedness.
In essence, think of subsets as smaller pieces of a larger puzzle. They maintain a relationship and share properties with the larger set they belong to.

Topology

Topology is a branch of mathematics that studies the properties of a space that are preserved under continuous transformations. Unlike other areas of geometry, topology is more concerned with the qualitative aspects of objects rather than their quantitative aspects, such as size or shape.
When we talk about a topological space, like \((S, \rho)\), we are referring to a set \( S \) paired with a collection of open sets \( \rho \), which together define a sort of "geometric" structure that allows us to explore concepts of continuity, connectedness, and boundaries.
Topological properties, including connectedness, are properties that a space possesses regardless of how it might be bent, stretched, or twisted, provided no tearing or gluing occurs. In our proof exercise, we're exploring how the topology of a set affects the connectedness of its subsets, specifically how a subset connected in the larger space \( S \) retains this property within any larger subset \( A \) containing it.

Open Sets

Open sets are fundamental to the study of topology. They are the building blocks of topological spaces, as they allow us to define continuity, connectedness, and convergence in a more abstract context.
Mathematically, an open set within a topological space \((S, \rho)\) is a set from the collection \( \rho \) that satisfies specific conditions: inclusively, any point \( x \) in the open set possesses a neighborhood entirely contained within the set. This condition provides a sense of "freedom" around each point, differentiating open sets from merely being subsets.
Understanding open sets is crucial when investigating connectedness. In our example, we needed to demonstrate that if a subset \( B \) cannot be divided into two disjoint open sets in a larger space \( S \), the same holds true within any encompassing subset \( A \) of \( S \). Thus, open sets allow us to explore the foundation of topological ideas, connecting abstract concepts with tangible examples.