Mathematical Logic (Part II) January 16, 2015
1. Let R be set of real numbers.
(a) Define a nontrivial topology τ on R which does not satisfy the separation axiom T2. (b) Is the interval (a, b) open in the topology τ ?
Solution 1: (a) We can use the usual total ordering on R to define the Alexandrov topology.
A subset of R is open iff it is an upper set. R and the empty set are upper sets. The other ones are the intervals (a, +∞) and [a, +∞). If a < b then every upper interval containing a contains b. This means that a and b cannot be T2-separated.
(b) The interval (a, b) is not open because is not an upper set.
Solution 2: We can define the topology τ = {∅, R, {x ∈ R : x > 0}}. The reals 1 and 2, for example, cannot be T2-separated. The interval (a, b) does not belong to τ .
2. Provide an example of a connected space and an example of a non-connected space.
Solution: The Sierpinski space 2 = {0, 1} (determined by 0 < 1) is a connected space.
There is only one nontrivial open set {1}.
The space R with the topology τ = {∅, R, {x ∈ R : x > 0}, {x ∈ R : x ≤ 0}} is not connected: R = {x ∈ R : x > 0} ∪ {x ∈ R : x ≤ 0} and {x ∈ R : x > 0} ∩ {x ∈ R : x ≤ 0} = ∅.
3. Let 2 = {0, 1} be the Sierpinski space (determined by 0 < 1). Consider the space 2 × 2 with the product topology. Does the space 2 × 2 admit a subset which is clopen (that is, open and closed)?
Solution: The topology of 2 × 2 is generated by:
{1} × {1} = {(1, 1)}
{1} × {0, 1} = {(1, 0), (1, 1)}
{0, 1} × {1} = {(0, 1), (1, 1)}.
Another open set is {(1, 0), (1, 1)} ∪ {(0, 1), (1, 1)} = {(0, 1), (1, 1), (1, 0)}.
Every open set contains the pair (1, 1). Then there exist no clopens.
4. Let R × R be the real plane with the euclidean topology. Determine the interior of the following sets (interior = the greatest open set included within the set) :
(a) A = {(x, y) : y > x}
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(b) B = {(x, y) : x = y}
(c) C = {(x, y) : x < 0, y ≥ 5}.
Solution: A is open. Then the interior of A is A. B is a line. A line cannot contains a bidimensional ball. Then the interior of B is the empty set. The interior of C is the set {(x, y) : x < 0, y > 5}.
5. Let X be a topological space, 2 = {0, 1} be the Sierpinski space and f : X → 2 be a function. Characterize when f is continuous.
Solution: f is continuous iff the inverse image of a nontrivial open set is open. The only nontrivial open set of the Sierpinski space is {1}. Then f is continuous iff {x ∈ X : f (x) = 1} is an open set.
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