In the realm of topology, a subset refers to a portion taken from a set, whereas a subspace denotes a subset that acquires the topological attributes of its parent set.
To elaborate, a subset is essentially an assembly of elements originating from a set, while a subspace signifies a subset that not only contains elements but also mirrors the topological characteristics exhibited by the original set.
For instance, the set encompassing all real numbers establishes a topological space. On the other hand, the set of rational numbers stands as a subset within the realm of real numbers, yet it lacks the status of a subspace due to its inability to mirror the topological traits of real numbers. Rational numbers do not qualify as open sets within real numbers, yet they do hold that status within the realm of rational numbers.
For a subspace to be valid, it must fulfill these conditions:
• It must essentially exist as a subset of its parent set
• It should uphold closure under topological actions, meaning that if A and B are open sets within the subspace, then their union and intersection must also emerge as open sets within the same subspace.
• It ought to maintain closure under scalar multiplication. This indicates that if an element x is present in the subspace and 'c' denotes a real number, the product of 'c' and 'x' should also exist within the subspace.
A subset that meets all these criteria earns the title of a topological subspace.
Generally, a subset may or may not satisfy these conditions to become a subspace. A subspace will always possess the characteristics of a subset, although the reverse doesn't necessarily hold true.