Final Test in MAT 410: Introduction to Topology Answers to the ... - Gap

7. R with the usual topology is a compact topological space.
true ( ) false ( X )
The open cover R = ∪ n∈Z ]n, n + 2[ does not possess a finite subcover.
8. R with the Zariski topology is a compact topological space.
true ( X ) false ( )
See Exercise 29.
9. R with the usual topology is a connected topological space.
true ( X ) false ( )
The only sets in R which are both open and closed are ∅ and R.
10. R with the Zariski topology is a connected topological space.
true ( X ) false ( )
No subset of R is both finite and has a finite complement – so the above holds also
here.
11. All Hausdorff spaces with countably many points are compact.
true ( ) false ( X )
Counterexample: Z with the discrete topology. – Then the open cover consisting of
the 1-element subsets possesses no finite subcover.
12. In a compact metric space, every sequence of points has a convergent subsequence.
true ( X ) false ( )
See Theorem 7.22 in the script.
13. Finite topological spaces are always connected.
true ( ) false ( X )
Counterexample: discrete topological space with at least 2 points.
14. Finite topological spaces are never connected.
true ( ) false ( X )
Counterexample: any finite set with the trivial topology.
15. There are Hausdorff spaces which are totally disconnected.
true ( X ) false ( )
Example: discrete topological spaces.
16. Let Z be endowed with the topology where the open sets are the set-theoretic unions
of residue classes. Then f : Z → Z, n ↦→ n + (−1) n is an homeomorphism.
true ( X ) false ( )
The function f maps unions of residue classes to unions of residue classes, so it is
an homeomorphism.
17. The set S ⊂ R n (n ∈ N) of zeros of a polynomial P ∈ R[x 1 , . . . , x n ] is always
bounded, and its diameter is bounded above by the product of the coefficients of P .
true ( ) false ( X )
Counterexample: the set of zeros of the zero polynomial is R n , and hence unbounded.
18. Let A be a bounded subset of R 3 , and let B ⊂ R 3 be a superset of A such that
B \ A is finite. Then the diameter of B is the same as the diameter of A.
true ( ) false ( X )
B may contain a point ‘far away’ from A.
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