Midterm Test in MAT 410: Introduction to Topology
Answers to the Test Questions (Variation 1)
Stefan Kohl
Question 1: Give the definition of a topological space. (3 credits)
Answer: Atopological space (X, τ ) is a pair consisting of a set Xand a collection τof subsets of X
such that the following hold:
• {∅, X} ⊂ τ.
•τis closed under taking arbitrary unions and finite intersections.
The sets in τare termed open sets.
Question 2: Give the definition of a metric space. (3 credits)
Answer: Ametric space (X, d) is a pair consisting of a set Xand a distance function d:X×X→R+
0
such that the following hold:
•d(x, y)=0⇔x=y.
• ∀x, y ∈X d(x, y) = d(y, x) (Symmetry).
• ∀x, y, z ∈X d(x, y) + d(y , z)>d(x, z) (Triangle Inequality).
Question 3: Let X:= {1,2,3,4}be a set of cardinality 4, and define a topology on Xby letting the
open sets be {4},{1,2},{1,2,3}and {2,3,4}. Either prove that Xwith this collection of open sets
is a topological space, or list at least 4 violations of the axioms of a topological space. (4 credits)
Answer: 1. ∅is not open, 2. Xis not open, 3. the union of the open sets {1,2}and {4}is not
open, and 4. the intersection of the open sets {1,2,3}and {2,3,4}is not open.
Question 4: Let X:= {1,2,3,4}be a set of cardinality 4, and let d:X×X→R+
0be the mapping
defined by d(1,1) = 0, d(1,2) = 1, d(1,3) = 6, d(1,4) = 2, d(2,1) = 2, d(2,2) = 0, d(2,3) = 4,
d(2,4) = 1, d(3,1) = 6, d(3,2) = 4, d(3,3) = 2, d(3,4) = 2, d(4,1) = 2, d(4,2) = 1, d(4,3) = 2 and
d(4,4) = 0. Either prove that (X, d) is a metric space, or list at least 4 violations of the axioms of a
metric space.
(4 credits)
Answer: 1. d(3,3) = 1 violates the axiom that a point has distance 0 from itself, 2. d(1,2) = 1 and
d(2,1) = 2 violate the axiom of symmetry, i.e. that the distance from ato bis always the same as
the distance from bto a, 3. d(1,4) = 2, d(4,3) = 2 and d(1,3) = 6 violate the triangle inequality,
and 4. d(2,4) = 1, d(4,3) = 2 and d(2,3) = 4 violate the triangle inequality as well.
Question 5: When is a topological space said to be a Hausdorff space? (2 credits)
Answer: Iff distinct points have disjoint neighbourhoods.
Question 6: Let X:= {1,2,3,4}. Give an example of a topology with which Xbecomes a Hausdorff
space. (2 credits)
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