2014 08 01 Products and Subspaces

8/11/2019 2014 08 01 Products and Subspaces

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Subspaces and products 1 / 14

Subspaces and products

2014-08-01 09:00
http://goforward/http://find/http://goback/


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Subspaces and products j Subspace topology 2 / 14

1 Subspace topology

2 Product topology

3 Exercises

4 Links


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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.


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Denition




is a topology on Y, called subspace topology.

Verify that fiY is indeed a topology on Y.


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Denition




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.


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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


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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


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Example

Subspace of the real line



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Example


Consider Y = [0; 1] X = R, where X has the standardtopology.



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Example



A base for the subspace topology on Y is the collectionof all sets of the form (a; b) \ Y.



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Example




They all are of one of the following forms: (a; b), [0; b),(a; 1], Y, ;.



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Example




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].



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p e p o j p e opo ogy 5 /

Example

Subspace and order topology not always the same



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j gy /

Example


Consider now Y = [0; 1) [ f2g X = R.



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Example



In the subspace topology, the set f2g is open.



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Example



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.



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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.



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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


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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 .



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Proof.



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Proof.

Denition (Product topology)

The topology just mentioned is called the product

topology on the Cartesian product X Y.



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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 .



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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.



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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.



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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.


P j ti


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Projections

Denition (Projections)The maps 1: X Y ! X and 2: X Y ! Y are calledprojections of X Y.


P j ti


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Projections


Inverse images of projections

If U is open in X, then `11 (U) =


P j ti


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Projections



If U is open in X, then `11 (U) =


Projections


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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.


Projections


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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


Subbase of the product


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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 .


Subspace and products


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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 .




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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


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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).



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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.



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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


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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_topology


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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

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2014 08 01 Products and Subspaces

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