MTH634 Final Term Slides Modules 86-183 by Fira Neehson

Def: A map Gort f :(X, Tx)

> (Y, Ty)

is called a closed map iff the image of each closed subset of X is closed in Y.

PF Def: A map

f :(X, Tx)

+ (Y, Ty)

is called a closed map iff the image of each closed subset of X is closed in Y. Y =) qc

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Topology

Projection Mapping A, AS

rA

Projection Mapping

Projection mapping is an example of mapping that is not closed in general.

But let us see first " What is projection map?"

aA

A

Projection Mapping

Projection Map: Let (X. Tx) and (Y. Ty) be two topological spaces. The

mere

Prods Sob · ™m (XxY · x.y) · (X.Tx) · (x.y)

x

T2:(Xx Y.Txxv) 7 (Y.Ty) · (x.y) > · y

ar called projection maps, where 7x,y is product topology

on X x Y. 7, is first projection and 72 is second projection.

Wa · Projection Mapping · Fact:

Projection map may or may not be closed.

Justification:

Consider the first projection

Wy: (R x R. Te xr) > (R. 7.)

The set A = {(x.y)|

y =tanx

, =? < x < 5} is closed in y R x R but

—Tt

m(A) = (=.5) · Wa · Projection Mapping · Fact:

Projection map may or may not be closed.

Justification:

Consider the first projection

J

m

The setA = {(x.y)|

y=tanx , | < x < $} is closed in

R x R but

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ca rw

q

(R x R.T.-n)

Tu)

y · Wa · Topology

Continuity and Open

Mapping

Continuity and Open Mapping

We have studied

> Continuous Mapping

>» Open Mapping

Questions:

1. Is a continuous map an

open map?

2. Is an open map a continuous map? A

f t

Continuity and Open Mapping

Example 1:

Let X = {a.b.c,d} with Tx = {0.{b.c}, {a,b,c}. X}

and

Consider the following map

f:xX +Y

defined as f(x) = 2 for all x € X.

Continuity and Open Mapping

Example 1:

Lod

J,

Let X = {a.b.c.d} with Tx = {0.({b. c}} {a,b,.c}.X}

and

Consider the following map

ti 4); {1

fe *

(\Ne es

defined as f(x) = 2 for all x € X.

Y

(4 Y

whee),

Wa

Continuity and Open Mapping

Example 2: Let X = R with usual topology and Y = R with lower limit

topology.

Consider the identity map

[a \) Clk, · PX · Prey fablcn,

xX x

t

iS

hot

Cont.

(a,b) Eh. · f(s) · 2 (c \,) é\ · Wa · Continuity and Open Mapping

Answers:

1. Continuous map may or

may not be open map.

2. Open map may or may not be continuous map.

ei

Topology

Openness and Restriction Mapping AA

t

I Recall the restriction mapping and the fact about the continuity of restriction

mapping. Now the question is

" Can we prove the same kind of result for restriction of an

open mapping?". f Example:

Let X = {a.b.c.d} with

Tx = {W. {a}. {a. b}. {a. b,c}. X} and

Consider the map f : X Y "4

Let A = {b.c.d}

Cc X with relative topology

Ta — {. {b}. {b. cc}. A}. Now the restriction map

f,

A —» Y “4 A

f4(b) = 2. fa(c) = 3. fa(d) = 2

Wa

Now f is an open map but f& is not an open map. Because

{5} is an open subset of A but

fal{b})= {2}

CY

NorQpen

Remark:

Restriction of an open map may or may not be an open

map.

KHALIL SAHIR

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Topology · ind · fig

owe. One

f

1%

yr rd

A mapf :X

—> Y is calleda

homeomorphism between two topological spaces

Y

(X.Tx) and (Y. Ty) iff

1. f is bijective.

A map f:X

> Y iscalleda

th Yo Y homeomorphism between ope” two topological spaces

Y

if

(X, Tx) and (Y. Ty) iff · NE · k

1. f is bijective.

pt~

f

d Gq

2. f is continuous. fh yc \ A map f :X > Y is called a is

Y

open homeomorphism between two topological spaces

c · (X. 7x) and (Y. Ty) iff · oP

1. f is bijective.

Y

2. f is continuous. 3. FY 4X is

continuous.

Two topological spaces

(X. Tx) and (Y.7y) ar

called homeomorphic iff

there exists a homeomorphism f

: X > Y.

We also say such spaces

topologically equivalent

spaces.

Yul

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a

Topology

Examples: NA»

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