Mathematical Logic (Part II) December 19, 2014
1. Let X = {a, b, c} be a set with three elements.
(a) Define two topologies τ1 and τ2 on X, which are distinct from the discrete and indiscrete topology.
Solution: For example, the following are nontrivial topologies on X: τ1 = {∅, X, {a}}
and τ2 = {∅, X, {b}, {b, c}}.
(b) What separation axioms hold in the space (X, τ1)? What separation axioms hold in the space (X, τ2)?
Solution:
(τ1): The elements b and c have the same neighbourhoods in the topology τ1. Then b ≤ c and c ≤ b in the specialization order of τ1.
(τ2): The topology τ2satisfies the separation axiom T0: we have a ≤ b ≤ c. We have a neighbourhoud {b, c} of c not containing a; a neighbourhoud {b} of b containing neither c nor a. The topology τ2 does not satisfy the separation axiom T1 and the other ones, because the partial ordering a ≤ b ≤ c is not trivial.
2. Let N be set of natural numbers with the usual partial ordering. Consider the Alexandrov topology on N. Is this space connected?
Solution: A space X is connected if there exist no open sets U and V such that U, V 6= ∅, U ∪ V = X and U ∩ V = ∅
The space N with the Alexandrov topology is connected. The open sets are the infinite intervals [k) = {x ∈ N : x ≥ k} and [k) ∩ [j) = [max{k, j}).
3. Let 2 = {0, 1} be the Sierpinski space (determined by 0 < 1). Consider the space 2 × 2 with the product topology. Enumerate all the open sets of 2 × 2.
Solution: The Cartesian product of two sets A and B is defined as follows: A × B = {(a, b) : a ∈ A, b ∈ B}. The element (a, b) is an ordered pair with first element a and second element b.
A base of open sets in the space 2 × 2 are the sets U × V , where U, V are opens in 2.
Then we have:
∅
2 × 2 = {(0, 0), (0, 1), (1, 0), (1, 1)}
{1} × {1} = {(1, 1)}
{1} × 2 = {(1, 0), (1, 1)}
2 × {1} = {(0, 1), (1, 1)},
(2 × {1}) ∪ ({1} × 2) = {(0, 1), (1, 1), (1, 0)}.
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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 = 3x}
(b) B = {(x, y) : x2+ y2 ≤ 3}
(c) C = {(3, 2)}.
Solution: int(A) = int(C) = ∅ and int(B) = {(x, y) : x2 + y2 < 3}.
A is a line (C is a point) in the plane and a line (point) does not contains any ball. B is a circle and any point (x, y) satisfying x2+ y2 < 3 admits a ball of center the point which is within the circle. The points satisfying x2+ y2 = 3 do not have this property.
5. Let R+ = {x ∈ R : x ≥ 0} be the real line with the euclidean topology and N be set of natural numbers with the Alexandrov topology. Is the function f : R+ → N, defined by f (x) = bxc (where bxc = max{k ∈ N : k ≤ x}) continuous?
Solution: f−1([3)) = {r ∈ R+ : r ≥ 3}. It is a closed set, not an open set! Then the function is not continuous.
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