M6140 Topology Exercises - 3rd Week (2020)
1 Separation Axioms
Exercise 1. Show that a disjoint union of T0 spaces is T0.
Exercise 2. Prove that a topological space is T0 iff any two distinct points have distinct closures.
Exercise 3. Show that a topological space X is T1 iff each singleton {x} in X is equal to the
intersection of all open neighborhoods of x.
Exercise 4. Prove that a topological space X is T2 iff each singleton {x} in X is equal to the
intersection of all closed neighborhoods of x.
Exercise 5. Show that a closed subspace of a T4 space is T4.
Exercise 6. Let F1, F2, F3 be a triple of closed sets in a T4 space X such that F1 ∩F2 ∩F3 = ∅. Show
that there exist open sets U1, U2, U3 in X such that F1 ⊆ U1, F2 ⊆ U2, F3 ⊆ U3 and U1 ∩ U2 ∩ U3 = ∅.
Exercise 7. A topological space is called irreducible if it cannot be written as a union of two proper
closed sets. Furthermore, a topological space is called sober if each irreducible closed set in it is a
closure of a unique point.
(i) Consider N with the cofinite topology. Prove that this space is not sober. This gives us an
example of a T1 space that isn’t sober.
(ii) Show that every T2 space is sober.
(iii) Show that every sober space is T0.
2 Compactness
Exercise 8. Show that a subspace A of a topological space X is compact iff for each collection
{Ui | i ∈ I} of open sets in X satisfying A ⊆ i∈I Ui there exists a finite subcollection {Uj | j ∈ J}
such that A ⊆ j∈J Uj.
Exercise 9. Prove that a finite union of compact subspaces of a topological space is a compact
subspace.
Exercise 10. Let A, B be disjoint compact subspaces of a Hausdorff topological space X. Show that
there exist disjoint open sets U, V in X such that A ⊆ U and B ⊆ V .
Exercise 11. A collection of subsets of a topological space is said to have a finite intersection property
if each finite subcollection has a non-empty intersection. Prove that a topological space X is compact
iff each collection of closed sets in X with finite intersection property has a non-empty intersection.
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Exercise 12. Show that a continuous map out of a compact space into a T2 space is closed and
proper. (A proper map is a continuous map such that a pre-image of a compact subspace is a compact
subspace.)
Exercise 13. (i) Suppose that f : X → Y is a continuous map between topological spaces whose
image is dense in Y and Y is T2. Prove that the cardinality of Y is at most |P(P(X))|.
(ii) Suppose that X is a topological space. By the previous part there exists a set I of isomorphism
classes of continous maps X → Y whose image is dense, and Y is compact and T2. For
each isomorphism class choose a representative fi : X → Yi. Consider the continous mapping
(fi)i∈I : X → i∈I Yi and let β(X) be the closure of its image. The space β(X) is called the
Stone- ˇCech compactification of X. Show that β(X) is compact and T2.
(iii) Suppose that f : X → Y is a continuous map between topological spaces such that Y is compact
and T2. Prove that f uniquely factorizes through the mapping X → β(X).
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