Subspace Topology on A: Calculate T_A

In summary, the subspace topology on A induced by the given topology T is Ta = { emptyset, {a}, {c}, {b,c}, {a,c}, {a,b,c}, {b,c} }.
  • #1
Shaggydog4242
5
0

Homework Statement



1. Let A= {a,b,c}. Calculate the subspace topology on A induced by the topology

T= { empty set, X,{a},{c,d},{b,c,e},{a,c,d},{a,b,c,e},{b,c,d,e},{c}, {a,c}} on X={a,b,c,d,e}.

Homework Equations



Given a topological space (X, T) and a subset S of X, the subspace topology on S is defined by

Ts= {S[itex]\bigcap[/itex]U s.t. U[itex]\in[/itex]T}

The Attempt at a Solution



Would it just be Ta = { emptyset, {a}, {c}, {b,c}, {a,c}, {a,b,c}, {b,c} }?
 
Physics news on Phys.org
  • #2
Shaggydog4242 said:

Homework Statement



1. Let A= {a,b,c}. Calculate the subspace topology on A induced by the topology

T= { empty set, X,{a},{c,d},{b,c,e},{a,c,d},{a,b,c,e},{b,c,d,e},{c}, {a,c}} on X={a,b,c,d,e}.


Homework Equations



Given a topological space (X, T) and a subset S of X, the subspace topology on S is defined by

Ts= {S[itex]\bigcap[/itex]U s.t. U[itex]\in[/itex]T}

The Attempt at a Solution



Would it just be Ta = { emptyset, {a}, {c}, {b,c}, {a,c}, {a,b,c}, {b,c} }?

Seems ok to me.
 

FAQ: Subspace Topology on A: Calculate T_A

What is the definition of subspace topology?

The subspace topology on a set A is a topology that is induced by a subset of A. This means that the open sets in the subspace topology are the intersections of open sets in the original topology on A with A itself.

How do you calculate the subspace topology on a set A?

The subspace topology on A can be calculated by taking the collection of all subsets of A that are open in the original topology on A. These subsets will form the open sets in the subspace topology.

What is the purpose of subspace topology?

The purpose of subspace topology is to study the topological properties of a subset of a space. It allows us to focus on a smaller, more manageable space while still preserving the important topological properties of the larger space.

How does subspace topology relate to the original topology on A?

The subspace topology is a subset of the original topology on A, meaning that all the open sets in the subspace topology are also open in the original topology. This allows us to use the tools and concepts from the original topology to study the subspace.

Are there any special cases where the subspace topology is different from the original topology?

Yes, there are cases where the subspace topology can be different from the original topology on A. This can happen when the subset A is not a proper subset of the original space, or when the subspace has a different cardinality than the original space. In these cases, the subspace topology may not have the same open sets as the original topology.

Similar threads

Path-connectedness for finite topological spaces
Replies
3
Views
2K
Relation between components and path-components of ##X##
Replies
4
Views
2K
Determine all of the open sets in given product topology
Replies
3
Views
1K
Writing some ZF axioms with FOL symbols
Replies
2
Views
1K
The union of three subspaces of V is a subspace of V
Replies
4
Views
2K
"Show B is closed if and only if...." (More Topology....)
Replies
4
Views
1K
Subspace topology and Closed Sets
Replies
6
Views
2K
I Fiber bundle homeomorphism with the fiber
2
Replies
43
Views
3K
Can a subbasis help determine which topology is finer?
Replies
3
Views
2K
[LinAlg] Set Notation, Subspaces, Bases, Ranges, & Kernels
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top