Norms 5 - 7 Flashcards
Define a topology t on a set T
What is a topological space
The pair (T, t)
What are the open sets of the discrete metric
all subsets of X
What are the open sets of the indiscrete topology
T and the empty set
What are the open sets of the Zariski topology
X, empty set and all sets with finite complements
Define coarser and finer for two topologies t1 and t2
If t1 is a subset of t2 we say t1 is coarser than t2, and t2 is larger than t1
Give the three conditions for a subset of a topological space to be closed
T and empty set are closed
finite union of closed sets are closed
arbitrary intersections of closed sets are closed
Define a basis for a topology t on T
State the two conditions for B, any basis of t
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
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
Define a sub-basis for a topology t on T
Define the subspace topology on S
Draw the diagram linking (X,d), (S, ds), t and ts
Define the topological product of T1 and T2
Define a neighbourhood of x in T
Define the closure
Define the interior
Define the boundard dH of a set H
the set of all points x where every neighbourhood meets both H and its complement
What is the boundary of (a,b)
d(a,b) = d[a,b] = {a.b}
Define a limit point a set S
Define dense, nowhere dense and meagre
Lemma: A subset that’s closed in a closed subspace of a topological space is closed in the whole space
When is a topological space (T, t) metrisable?
When there is a metric d such that t consists of the open sets in (T, d)
Give the definition of convergence for a sequence ina topological space
Define a Hausdorff space
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
Lemma: In a Hausdorff space T, any sequence can have at most one limit
Give a necessary condition for a topological space to be metrisable
It must be Hausdorff
Define the topology of pointwise convergence
Define continuity between topological spaces
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
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
Define two projections on T1 x T2
The projection pij is continuous from T1 x T2 to Tj
If U1 subset of T1 is open, then (pi1)-1(U1) = U1 x T2 which is open
Lemma: If f,g: T to R are continuous then so is f + g
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
Define a homeomorphism between topological spaces
Define a topological invariant and state 4
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
