Upload
ultimouniverso
View
224
Download
0
Embed Size (px)
Citation preview
8/11/2019 2014 08 01 Products and Subspaces
1/38
Subspaces and products 1 / 14
Subspaces and products
2014-08-01 09:00
http://goforward/http://find/http://goback/8/11/2019 2014 08 01 Products and Subspaces
2/38
Subspaces and products j Subspace topology 2 / 14
1 Subspace topology
2 Product topology
3 Exercises
4 Links
8/11/2019 2014 08 01 Products and Subspaces
3/38
Subspaces and products j Subspace topology 2 / 14
Denition
Denition (Subspace topology)
Let (X; fi) be a topological space and Y X. Then
fiY = fY \ U j U 2 fig
is a topology on Y, called subspace topology.
8/11/2019 2014 08 01 Products and Subspaces
4/38
Subspaces and products j Subspace topology 2 / 14
Denition
Denition (Subspace topology)
Let (X; fi) be a topological space and Y X. Then
fiY = fY \ U j U 2 fig
is a topology on Y, called subspace topology.
Verify that fiY is indeed a topology on Y.
8/11/2019 2014 08 01 Products and Subspaces
5/38
Subspaces and products j Subspace topology 2 / 14
Denition
Denition (Subspace topology)
Let (X; fi) be a topological space and Y X. Then
fiY = fY \ U j U 2 fig
is a topology on Y
, called subspace topology
.
Verify that fiY is indeed a topology on Y.
Open sets relative to Y
If Y is a subspace of X, and U Y, it could be that U isopen in Y but not open in X.
8/11/2019 2014 08 01 Products and Subspaces
6/38
Subspaces and products j Subspace topology 3 / 14
Base of the subspace topology
Lemma
Let B be a base for the topology of X. Then
BY = fB\ Y j B 2 Bg
is a base for the subspace topology on Y.
S b d d t j S b t l 3 / 14
8/11/2019 2014 08 01 Products and Subspaces
7/38
Subspaces and products j Subspace topology 3 / 14
Base of the subspace topology
Lemma
Let B be a base for the topology of X. Then
BY = fB\ Y j B 2 Bg
is a base for the subspace topology on Y.
Proof.
S b d d t j S b t l 4 / 14
8/11/2019 2014 08 01 Products and Subspaces
8/38
Subspaces and products j Subspace topology 4 / 14
Example
Subspace of the real line
Subspaces and products j Subspace topology 4 / 14
8/11/2019 2014 08 01 Products and Subspaces
9/38
Subspaces and products j Subspace topology 4 / 14
Example
Subspace of the real line
Consider Y = [0; 1] X = R, where X has the standardtopology.
Subspaces and products j Subspace topology 4 / 14
8/11/2019 2014 08 01 Products and Subspaces
10/38
Subspaces and products j Subspace topology 4 / 14
Example
Subspace of the real line
Consider Y = [0; 1] X = R, where X has the standardtopology.
A base for the subspace topology on Y is the collectionof all sets of the form (a; b) \ Y.
Subspaces and products j Subspace topology 4 / 14
8/11/2019 2014 08 01 Products and Subspaces
11/38
Subspaces and products j Subspace topology 4 / 14
Example
Subspace of the real line
Consider Y = [0; 1] X = R, where X has the standardtopology.
A base for the subspace topology on Y is the collectionof all sets of the form (a; b) \ Y.
They all are of one of the following forms: (a; b), [0; b),(a; 1], Y, ;.
Subspaces and products j Subspace topology 4 / 14
8/11/2019 2014 08 01 Products and Subspaces
12/38
Subspaces and products j Subspace topology 4 / 14
Example
Subspace of the real line
Consider Y = [0; 1] X = R, where X has the standardtopology.
A base for the subspace topology on Y is the collectionof all sets of the form (a; b) \ Y.
They all are of one of the following forms: (a; b), [0; b),(a; 1], Y, ;.
These are precisely the basic elements of the ordertopology on [0; 1].
Subspaces and products j Subspace topology 5 / 14
8/11/2019 2014 08 01 Products and Subspaces
13/38
p e p o j p e opo ogy 5 /
Example
Subspace and order topology not always the same
Subspaces and products j Subspace topology 5 / 14
8/11/2019 2014 08 01 Products and Subspaces
14/38
j gy /
Example
Subspace and order topology not always the same
Consider now Y = [0; 1) [ f2g X = R.
Subspaces and products j Subspace topology 5 / 14
8/11/2019 2014 08 01 Products and Subspaces
15/38
Example
Subspace and order topology not always the same
Consider now Y = [0; 1) [ f2g X = R.
In the subspace topology, the set f2g is open.
Subspaces and products j Subspace topology 5 / 14
8/11/2019 2014 08 01 Products and Subspaces
16/38
Example
Subspace and order topology not always the same
Consider now Y = [0; 1) [ f2g X = R.
In the subspace topology, the set f2g is open.
But in the order topology on Y, considering that 2 is
max Y, any open set that contains 2 has to contain
other points.
Subspaces and products j Subspace topology 6 / 14
8/11/2019 2014 08 01 Products and Subspaces
17/38
Order topology in intervals
Lemma
If X is an ordered set that has the order topology, and
Y X is an interval, then the order topology and the
subspace topology on Y are the same.
Subspaces and products j Subspace topology 6 / 14
8/11/2019 2014 08 01 Products and Subspaces
18/38
Order topology in intervals
Lemma
If X is an ordered set that has the order topology, and
Y X is an interval, then the order topology and the
subspace topology on Y are the same.
Proof.
Exercise
Subspaces and products j Product topology 7 / 14
8/11/2019 2014 08 01 Products and Subspaces
19/38
A base for the product topology
Theorem (Base of the product topology)
Let X and Y be topological spaces. Let B be thecollection of all subsets of X Y of the form U V ,where U is open in X and V is open in Y . Then B is abase for a topology on X Y .
Subspaces and products j Product topology 7 / 14
8/11/2019 2014 08 01 Products and Subspaces
20/38
A base for the product topology
Theorem (Base of the product topology)
Let X and Y be topological spaces. Let B be thecollection of all subsets of X Y of the form U V ,where U is open in X and V is open in Y . Then B is abase for a topology on X Y .
Proof.
Subspaces and products j Product topology 7 / 14
8/11/2019 2014 08 01 Products and Subspaces
21/38
A base for the product topology
Theorem (Base of the product topology)
Let X and Y be topological spaces. Let B be thecollection of all subsets of X Y of the form U V ,where U is open in X and V is open in Y . Then B is abase for a topology on X Y .
Proof.
Denition (Product topology)
The topology just mentioned is called the product
topology on the Cartesian product X Y.
Subspaces and products j Product topology 8 / 14
8/11/2019 2014 08 01 Products and Subspaces
22/38
Product by a base
Theorem (Product topology given by a base)
Let X be a topological space with base B, and Y be aspace with basis C. Then the set
D = fU V j U 2 B; V 2 Cg
is a base for the product topology on X Y .
Subspaces and products j Product topology 8 / 14
8/11/2019 2014 08 01 Products and Subspaces
23/38
Product by a base
Theorem (Product topology given by a base)
Let X be a topological space with base B, and Y be aspace with basis C. Then the set
D = fU V j U 2 B; V 2 Cg
is a base for the product topology on X Y .
Proof.
Subspaces and products j Product topology 9 / 14
8/11/2019 2014 08 01 Products and Subspaces
24/38
Examples
Example (Product topology on R2)
A base for the product of two standard topologies on R is
the collection of all rectangles of the form (a; b) (c; d).It follows that the product topology is the same as the
metric topology on R.
Subspaces and products j Product topology 9 / 14
8/11/2019 2014 08 01 Products and Subspaces
25/38
Examples
Example (Product topology on R2)
A base for the product of two standard topologies on R is
the collection of all rectangles of the form (a; b) (c; d).It follows that the product topology is the same as the
metric topology on R.
Example (Product of Sierpinski spaces)
Describe the points and the open sets on the productX X, when X is a Sierpinski space.
Subspaces and products j Product topology 10 / 14
P j ti
8/11/2019 2014 08 01 Products and Subspaces
26/38
Projections
Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.
Subspaces and products j Product topology 10 / 14
P j ti
8/11/2019 2014 08 01 Products and Subspaces
27/38
Projections
Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.
Inverse images of projections
If U is open in X, then `11 (U) =
Subspaces and products j Product topology 10 / 14
P j ti
8/11/2019 2014 08 01 Products and Subspaces
28/38
Projections
Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.
Inverse images of projections
If U is open in X, then `11 (U) =
Subspaces and products j Product topology 10 / 14
Projections
8/11/2019 2014 08 01 Products and Subspaces
29/38
Projections
Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.
Inverse images of projections
If U is open in X, then `11 (U) = U Y, which is open in
X Y. Similarly, if V is open in Y, the set
`1
2 (V
) = X V, is open in X Y.
Subspaces and products j Product topology 10 / 14
Projections
8/11/2019 2014 08 01 Products and Subspaces
30/38
Projections
Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.
Inverse images of projections
If U is open in X, then `11 (U) = U Y, which is open in
X Y. Similarly, if V is open in Y, the set
`1
2 (V
) = X V, is open in X Y.
We have
`11 (U) \
`12 (V) = U V
Subspaces and products j Product topology 11 / 14
Subbase of the product
8/11/2019 2014 08 01 Products and Subspaces
31/38
Subbase of the product
Theorem (Subbase)
The collection S = f 11 (U)gU2fiX [ f 12 (V)gV2fiY is asubbase for the product topology on X Y .
Subspaces and products j Product topology 12 / 14
Subspace and products
8/11/2019 2014 08 01 Products and Subspaces
32/38
Subspace and products
Theorem (Subspace and product topology)
Let A; B be subspaces of X; Y respectively. Then the
product topology on A B is the same as the topology on
A B as subspace of the product space X Y .
Subspaces and products j Product topology 12 / 14
Subspace and products
8/11/2019 2014 08 01 Products and Subspaces
33/38
Subspace and products
Theorem (Subspace and product topology)
Let A; B be subspaces of X; Y respectively. Then the
product topology on A B is the same as the topology on
A B as subspace of the product space X Y .
Proof.
Exercise
Subspaces and products j Exercises 13 / 14
8/11/2019 2014 08 01 Products and Subspaces
34/38
Show that if X is a topological space, and A Y X,then the topology of A as a subspace of Y is the same as
a subspace of X (Y is a subspace of X).
Subspaces and products j Exercises 13 / 14
8/11/2019 2014 08 01 Products and Subspaces
35/38
Show that if X is a topological space, and A Y X,then the topology of A as a subspace of Y is the same as
a subspace of X (Y is a subspace of X).
A map f : X ! Y between topological space is said to beopen if f(U) is open for every U X which is an open set.Prove that the projection X: X Y ! X is an open set.
Subspaces and products j Exercises 13 / 14
8/11/2019 2014 08 01 Products and Subspaces
36/38
Show that if X is a topological space, and A Y X,then the topology of A as a subspace of Y is the same as
a subspace of X (Y is a subspace of X).
A map f : X ! Y between topological space is said to beopen if f(U) is open for every U X which is an open set.Prove that the projection X: X Y ! X is an open set.
Let I denote the closed interval [0; 1] R as a subspaceof the usual topology on R. Compare the product
topology on I I, the dictionary order topology on I I,
and the product Id I, where Id denotes discrete topology.
Subspaces and products j Links 14 / 14
8/11/2019 2014 08 01 Products and Subspaces
37/38
Subspace topology - Wikipedia, the free encyclopedia
Subspaces and products j Links 14 / 14
http://en.wikipedia.org/wiki/Subspace_topologyhttp://en.wikipedia.org/wiki/Product_topologyhttp://en.wikipedia.org/wiki/Subspace_topology8/11/2019 2014 08 01 Products and Subspaces
38/38
Subspace topology - Wikipedia, the free encyclopedia
Product topology - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Subspace_topologyhttp://en.wikipedia.org/wiki/Product_topologyhttp://en.wikipedia.org/wiki/Product_topologyhttp://en.wikipedia.org/wiki/Subspace_topology