Norms 5 - 7

Question 1

Define a topology t on a set T

Answer 1

Question 2

What is a topological space

Answer 2

The pair (T, t)

Question 3

What are the open sets of the discrete metric

Answer 3

all subsets of X

Question 4

What are the open sets of the indiscrete topology

Answer 4

T and the empty set

Question 5

What are the open sets of the Zariski topology

Answer 5

X, empty set and all sets with finite complements

Question 6

Define coarser and finer for two topologies t₁ and t₂

Answer 6

If t₁ is a subset of t₂ we say t₁ is coarser than t_2, and t₂ is larger than t₁

Question 7

Give the three conditions for a subset of a topological space to be closed

Answer 7

T and empty set are closed

finite union of closed sets are closed

arbitrary intersections of closed sets are closed

Question 8

Define a basis for a topology t on T

Answer 8

Question 9

State the two conditions for B, any basis of t

Answer 9

B1 - T is the union of sets from B

B2 - if B₁, B₂ are in B then B₁ intersect B₂ is the union of sets from B

Question 10

Let B be a collection of subsets of a set T that satisfy B1 and B2. Then there is a unique topology of T whose basis is B; it’s open sets are precisely the union of sets from B

Answer 10

Question 11

Define a sub-basis for a topology t on T

Answer 11

Question 12

Define the subspace topology on S

Answer 12

Question 13

Draw the diagram linking (X,d), (S, d_s), t and t_s

Answer 13

Question 14

Define the topological product of T₁ and T₂

Answer 14

Question 15

Define a neighbourhood of x in T

Answer 15

Question 16

Define the closure

Answer 16

Question 17

Define the interior

Answer 17

Question 18

Answer 18

Question 19

Define the boundard dH of a set H

Answer 19

the set of all points x where every neighbourhood meets both H and its complement

Question 20

What is the boundary of (a,b)

Answer 20

d(a,b) = d[a,b] = {a.b}

Question 21

Define a limit point a set S

Answer 21

Question 22

Define dense, nowhere dense and meagre

Answer 22

Question 23

Lemma: A subset that’s closed in a closed subspace of a topological space is closed in the whole space

Answer 23

Question 24

When is a topological space (T, t) metrisable?

Answer 24

When there is a metric d such that t consists of the open sets in (T, d)

Question 25

Give the definition of convergence for a sequence ina topological space

Answer 25

Question 26

Define a Hausdorff space

Answer 26

A topological space T is Hausdorff if for any x,y in T there exists disjoint open sets U, V with x in U and y in V

Question 27

Lemma: In a Hausdorff space T, any sequence can have at most one limit

Answer 27

Question 28

Give a necessary condition for a topological space to be metrisable

Answer 28

It must be Hausdorff

Question 29

Define the topology of pointwise convergence

Answer 29

Question 30

Answer 30

Question 31

Define continuity between topological spaces

Answer 31

Question 32

Answer 32

Question 33

Lemma: If T₁, T₂, T₃ are topological spaces and f: T₁ to T₂ and g:T₂ to T₃ are continuous, gf: T₁ to T₃ is continuous

Answer 33

U open in T₃ means g^-1(U) is open in T₂ by continuity of g. So (gf)^-1(U) = f^-1(g^-1(U)) is open in T₁ by continuity of f

Question 34

Define two projections on T₁ x T₂

Answer 34

Question 35

The projection pi_j is continuous from T₁ x T₂ to T_j

Answer 35

If U₁ subset of T₁ is open, then (pi₁)^-1(U₁) = U₁ x T₂ which is open

Question 36

Answer 36

Question 37

Lemma: If f,g: T to R are continuous then so is f + g

Answer 37

The function r: R² to R given by r(x,y) = x + y is continuous. The map x going to (f(x), g(x)) is continuous from T to R². So r(T(x)) = f(x) + g(x) is continuous

Question 38

Define a homeomorphism between topological spaces

Answer 38

Question 39

Define a topological invariant and state 4

Answer 39

A property of topological spaces preserved by homeomorphisms

T is finite

T is hausdorff

T is metrisable

every continuous real valued function on T is bounded