Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

10 Topology (2301631)
Exercise 1.14. Prove that a closed [open] set in R n (with the usual topology)
is a G δ -set [F σ -set].
Definition 1.15. Let τ and τ ′ be two topologies on a set X. We say that τ ′
is finer than τ (or equivalently, τ is coarser than τ ′ ) if τ ⊆ τ ′ .
Example 1.16. τ dis is finer than τ indis . What can we say about τ cf and τ cc ?
Exercise 1.17. Let X be a space and Y ⊆ X.
{U ∩ Y | U ∈ τ X } forms a topology on Y .
Prove that the collection
Definition 1.18. Let X be a space and Y ⊆ X. The subspace topology of Y
(in X) is defined to be
τ Y = {U ∩ Y | U ∈ τ X }.
In this case, we say that (Y, τ Y ) is a subspace of X.
Example 1.19. The subspace topology of Z (in R) is discrete.
Exercise 1.20. Let Y be a subspace of X and A ⊆ Y . Prove that A is closed
in Y iff A = C ∩ Y for some closed set C in X.
Theorem 1.21. Let Y be an open [closed] subspace of X. If A is open [closed]
in Y , then A is also open [closed] in X.
Proof. Let A be open [closed] in Y . Then A = B ∩ Y for some open [closed]
subset B of X. Since Y itself is open [closed] in X and the interesection is finite,
it follows that A is also open [closed] in X.
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