Question

Problem 5. Let $(Y, \mathcal{T})$ be a topological space, and let $A \subset X \subset Y$ be subsets. We can consider two topologies on $A$: the subspace topology $\mathcal{T}_A$ on $A \subset Y$ considered as a subset of $Y$, and the subspace topology $(\mathcal{T}_X)_A$ on $A \subset X$ considered a subset of $X$ (where $X$ is regarded as a topological space via its subspace topology $\mathcal{T}_X$ inherited from $Y)$. Prove that $\mathcal{T}_A = (\mathcal{T}_X)_A$

          Problem 5. Let $(Y, \mathcal{T})$ be a topological space, and let $A \subset X \subset Y$ be subsets.
We can consider two topologies on $A$: the subspace topology $\mathcal{T}_A$ on $A \subset Y$ considered as a subset
of $Y$, and the subspace topology $(\mathcal{T}_X)_A$ on $A \subset X$ considered a subset of $X$ (where $X$ is regarded
as a topological space via its subspace topology $\mathcal{T}_X$ inherited from $Y)$.
Prove that $\mathcal{T}_A = (\mathcal{T}_X)_A$

Problem 5. Let (Y, 𝒯) be a topological space, and let A ⊂ X ⊂ Y be subsets.
We can consider two topologies on A: the subspace topology 𝒯A on A ⊂ Y considered as a subset
of Y, and the subspace topology (𝒯X)A on A ⊂ X considered a subset of X (where X is regarded
as a topological space via its subspace topology 𝒯X inherited from Y).
Prove that 𝒯A = (𝒯X)A

Added by Victor W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Tim Thornhill

Step 1

This means that the open sets in Ta are of the form U ∩ A, where U is an open set in Y. Show more…

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Problem 5. Let (Y,T) be a topological space, and let A C X C Y be subsets. We can consider two topologies on A: the subspace topology Ta on A C Y considered as a subset of Y, and the subspace topology (Tx)A on A C X considered a subset of X (where X is regarded as a topological space via its subspace topology Tx inherited from Y). Prove that Ta = (Tx)A.