Download Continuity in topological spaces and topological invariance

Continuity in topological spaces and topological
invariance
Juliet Cooke, Thomas Wright
June 29, 2013
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Examples of Topological Spaces
Definition 1 (Topological Space). For a set X and a system of subsets of X,
τ , we say τ is a topology in X and (X, τ ) is a topological space if
Axiom 1. ∅ ∈ τ, X ∈ τ
Axiom 2 (Closure under arbitrary union).
(∀α ∈ I, Gα ∈ τ ) ⇒
[
Gα ∈ τ
α∈I
Axiom 3 (Closure under finite intersection).
(∀i ∈ {1, . . . , n}, Gi ∈ τ ) ⇒
n
\
Gi ∈ τ
i=1
Example 1 (Discrete Topology). (X, τ ) is the discrete topology on set X where
τ = P(X) = {A : A ⊆ X}.
This is the topology induced by the discrete metric on a set X. Recall the
discrete metric on X is defined,
0 if x = y
d0 (x, y) :=
1 if x 6= y
Now for any G ⊆ X and x ∈ G, we see that Bε (x) = {x} ⊆ G if ε < 1, so
every subset G of X must be open, as is required for the discrete topology.
Example 2 (Indiscrete Topology). (X, τ ) is the indiscrete topology on set X
where τ = {∅, X}.
Example 3 (Cofinite Topology). (X, τ ) is the cofinite topology on set X where
τ = {A ⊆ X : X \ A is finite}.
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Explorations into Continuity in Metric and
Topological Spaces
We are going to consider how to generalize the notion of continuity of functions
to functions of topological spaces. To do this we are first going to generalize the
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definition of continuity from real analysis to metric spaces and then attempt to
rewrite this definition purely in terms of open sets to get a definition that will
correctly generalize.
Definition 2 (Real Analysis Definition of Continuity). Let f : R → R be a
function. f is continuous at a ∈ R if
∀ε ∈ R+ , ∃δ ∈ R+ , ∀x ∈ R, |x − a| < δ ⇒ |f (x) − f (a)| < ε
Recall that d (x, y) = |x − y| is the standard Euclidean metric for R. So
it would be simple to generalize this definition of continuity to metric spaces
by replacing this distance function d (x, y) with the more general definition of
distance given by the metric of a metric space. This is what we shall now do.
Definition 3 (Continuity in Metric Spaces). Let f : X → Y be a function
between metric spaces (X, dX ) and (Y, dY ). f is continuous at a ∈ X if
∀ε ∈ R+, ∃δ ∈ R+ , ∀x ∈ X, dX (x, a) < δ ⇒ dY (f (x), f (a)) < ε
There is a slight subtlety that the distance in the domain and image may be
different.
Now we will try to rewrite this definition purely in terms of open set but
first we will recall what we mean by an open set in a metric space.
Definition 4 (Open Ball). Let (X, d) be a metric space. The ball of radius
r ∈ R+
0 centered at point a ∈ X, Br (a) is defined to be
Br (a) = {x ∈ X : d(x, a) < r}
Definition 5 (Open Set in a Metric Space). Let (X, d) be a metric space with
subset A. A is d-open (usually written simply as open) in X if
∀a ∈ A, ∃r ∈ R+
0 , Br (a) ⊆ A
By looking at the definition of an open ball we can see that
d (x, a) ⇔ x ∈ Bε (a)
d (f (x), f (a)) ⇔ f (x) ∈ Bε (f (a))
So we can rewrite write the definition of continuity as,
f is continuous at a ∈ X ⇔ ∀ε ∈ R+ , ∃δ ∈ R+ , ∀x ∈ Bε (a) , f (x) ∈ Bδ (f (x))
By the definition of image:
(∀x ∈ Bε (a) , f (x) ∈ Bδ (f (x))) ⇔ f (Bε (a)) ⊆ Bε (a)
Hence we now have an equivalent definition of continuity:
Definition 6 (Continuity in Metric Spaces with respect to Open Balls). Let
f : X → Y be a function between metric spaces (X, dX ) and (Y, dY ). f is
continuous at a ∈ X if:
∀ ∈ R+ , ∃δ ∈ R+ , f (Bδ (a)) ⊆ Bε (f (a))
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Now, in expressing our definition in terms of open balls, we are one step closer
to generalizing our notion of continuity to a topological space. The definition of
an open ball is however dependent on our notion of distance; in order to fully
liberate our definition from metric space, we must instead consider an open
neighbourhood:
Definition 7 (Neighbourhood). A neighbourhood G of a point x ∈ X is any
set G ∈ τ such that x ∈ G (that is, any open set G in our topology containing
x).
Lemma 1. Let (X, d) be a metric space. Then Br (x) is an open neighbourhood
of x.
Proof. Firstly, x ∈ Br (x) as d (x, x) = 0 < r. Now, let y ∈ Br (x) and s :=
r − d (x, y) > 0. So for z ∈ Bs (y),
d (x, z) ≤ d (x, y) + d (y, z) < d (x, y) + s = r
Therefore Bs (y) ⊆ Br (x) proving that Br (x) is open.
Theorem 2 (Equivalence of definitions of Continuity for Metric Spaces). Let
(X, dX ), (Y, dY ) be metric spaces, and f : X → Y a function between them.
Then f is continuous at a point a ∈ X if and only if for every neighbourhood
B of f (a), there exists a neighbourhood A of a such that f (A) ⊆ B.
Proof. Assume f is continuous, and let B be an open neighbourhood of f (a). As
B is open and f (a) ∈ B, we have that ∃ε ∈ R+ , Bε (f (a)) ⊆ B. By continuity
of f , ∃δ ∈ R+ , f (Bδ (a)) ⊆ Bε (a) ⊆ B. By Lemma 1, Bδ (a) is an open
neighbourhood of a as required.
Conversely, for ε ∈ R+ , Bε (f (a)) is an open neighbourhood of f (a) by
Lemma 1. Hence there exists an open neighbourhood B of a such that f (B) ⊆
Bε (f (a)). As B is open and a ∈ B, there exists δ ∈ R+ such that Bδ (a) ⊆ B
therefore f (Bδ (a)) ⊆ f (B) ⊆ Bε (a), proving the continuity of f .
We now have an equivalent definition of continuity in metric spaces which
is independent of our notion of distance, requiring us only to consider open
neighbourhoods of a point. This allows us to consider continuity on an arbitrary
topological space.
Definition 8 (Continuity of functions in topological spaces). Let (X, τ ), (Y, υ)
be topological spaces, and f : X → Y be a function between them. Then f is
continuous at a point a ∈ X if for every neighbourhood B ∈ υ of f (a), there
exists a neighbourhood A ∈ τ of a such that f (A) ⊆ B.
Proposition 3 (Mapping to the indiscrete topology). Let (X, τ ) be a topological space and Y be a set equipped with the indiscete topology, υ = (∅, Y ).
Then any function f : X → Y is continuous.
Proof. If X = ∅, f is trivially continuous as we need consider no points.
If X 6= ∅, we may pick x ∈ X. Y ∈ υ is the only neighbourhood f (x). X is
a neighbourhood of x such that f (A) ⊆ Y , proving the continuity of f .
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Definition 9 (Everywhere continuous functions). Let (X, τ ), (Y, υ) be topological spaces and f : X → Y be a map between them. Then f is everywhere
continuous on X if and only if ∀U ∈ υ, f −1 (U ) ∈ τ .
Theorem 4. f is everywhere continuous on X if and only if f is continuous at
every point x ∈ X.
Proof. Assume that f is continuous at every point in X and let U ∈ υ. If
f −1 (U ) = ∅, it is open. Otherwise, we may pick x ∈ f −1 (U ), and so f (x) ∈ U .
Since f is continuous at x and U is a neighbourhood of f (x), we have that there
exists a neighbourhood V ∈ υ such that U ⊆ V , proving the preimage is open.
Conversely, assume that f is everywhere continuous on X. Let x ∈ X and
U be a neighbourhood of f (x). We need to find an open neighbourhood V ∈ τ
of x such that f (V ) ⊆ U . We see that V = f −1 (U ) is such a neighbourhood,
since openness follow from our assumption f is everywhere continuous, and we
know that f (f −1 (U )) ⊆ U .
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An introduction to topological properties
In this section we are going to define what in means for topological spaces to
be ‘essentially the same’. This is analogous to groups or vector spaces being
isomorphic. Unsurprisingly we use continuity to do this.
Definition 10 (Homeomorphisms between Topological Spaces). A homeomorphsim
between topological spaces X and Y is a bijective map f : X → Y such that
both f and f −1 are continuous.
If there exists a homeomorphism between two topological spaces they are
said to be homeomorphic or topologically equivalent. Topological equivalence
is an equivalence relation.
Properties of a topological space that are preserved under homeomorphism
are said to be topolocially invariant or simply topological properties.
Examples of topological properties include:
• Hausdorff property
• Compactness
• Connectedness
• Path Connectedness
All of which should be covered in future weeks.
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