Lecture 12 : Topological Spaces

Transcription

1 Leture 12 : Topologil Spes 1 Topologil Spes Topology generlizes notion of distne nd loseness et. Definition 1.1. A topology on set X is olletion T of susets of X hving the following properties. 1. nd X re in T. 2. The union of the elements of ny su-olletion of T is in T. 3. The intersetion of elements of ny finite su-olletion of T is in T. A set X for whih topology T hs een speified is lled topologil spe, denoted (X, T). Definition 1.2. If X is topologil spe with topology T, we sy tht suset U of X is n open set of X if U elongs to the olletion T. Exmple 1.3. Let X e ny set. 1. Let X = {,, }. Different olletions of susets of X re shown in the Figure 1. First nine of these olletions re topologies nd lst two re not. 2. Power set P(X) is topology lled the disrete topology. 3. Colletion T = {, X} is topology lled the indisrete topology or the trivil topology. 4. We define finite omplement topology on X s We will show T f is topology. T f = {U X : X\U is finite or X\U = X}. 1

2 T = {φ, X} T = {φ, {}, {, }, X} T = {φ, {, }, {}, {, }, X} T = {φ, {}, X} T = {φ, {}, {, }, X} T = {φ, {, }, {}, {, }, {}, X} T = {φ, {, }, X} T = {φ, {}, {, }, {}, X} T = {φ, {}, {, }, {}, {, }, {}, {, }, X} Figure 1: Different olletions of susets of X = {,, }. Lst two susets re not topologies. Memership of nd X. Sine X\ = X, we hve T f. Similrly, X T f sine X\X = is finite. Closure under ritrry union. Let {U i : i I} e n indexed fmily of elements in T f, with their union j I U j denoted y U. If U i = for ll i I, then there is nothing to show. Else, there exists i I suh tht X\U i is finite. Notie tht X\U = X \ j I U j = j I(X \ U j ) (X \ U i ). Sine X \ U i is finite, so is X\U. It follows tht union U T f. Closure under finite intersetion. For finite index set F, onsider {U i : i F } non-empty elements of T f, with their intersetion n i=1u i denoted 2

3 y V. We notie tht X\V = X\ n U i = i=1 n (X\U i ). i=1 Sine finite union of finite sets is finite, V T f. 5. We define o-ountle topology on X s T = {U X : X\U ountle or X\U = X}. Definition 1.4. Let T, T e topologies on set X. We sy tht topology T is 1. finer thn T, if T T, 2. stritly finer thn T, if T T, 3. is omprle to T, if either T T or T T. We sy T is orser or stritly orser in ove two ses, respetively. Definition 1.5. If T T, then we sy tht T is lrger thn T, nd T is smller thn T. Remrk 1. Topologies re not lwys omprle. In Figure 1, topology in first row nd first olumn is orser thn ll topologies, nd topology in third row nd third olumn is finer thn ll topologies. Topology in the seond row nd seond olumn is not omprle, to other two topologies in the third row. 2 Bsis for Topology Speifying whole olletion of open sets is prohiitive t times. One n speify smller olletion of susets of X nd define topology using them. Definition 2.1. If X is set, sis for topology on X is olletion B of susets of X, lled sis elements, suh tht 1. for ll x X, there exists B B suh tht x B, 2. for B 1, B 2 B nd x B 1 B 2, there exists B 3 B suh tht x B 3 B 1 B 2. 3

4 If B stisfies these two onditions, we define the topology T generted y B s follows. A suset U of X is lled open in X, if for eh x U, there exists B B suh tht x B U. Remrk 2. Oserve tht B T. Lemm 2.2. Colletion T generted y sis B is topology on X. Proof. Let T e the olletion of susets of X generted y the sis B on X. Memership of nd X. Let U e n empty set, in this se U vuously elongs to T. On the other hnd, for ll x X, there exists B suh tht x B X y definition of sis B. Closure under ritrry unions. Consider n indexed fmily of sets {U i T : i I}, nd define U = i I U i. For eh x U, there exists index i I suh tht x U i. Sine U i T, for eh x U i there exists sis element B x suh tht x B x U i U. Hene, U T y definition. Closure under finite intersetions. It suffies to show tht intersetion of two elements of T elongs to T. To this end, onsider U 1, U 2 T. Let x U 1 U 2, y definition we n hoose sis elements B 1 nd B 2 suh tht x B 1 U 1 nd x B 2 U 2. Using the seond ondition in the definition of the sis we n hoose sis element B 3 suh tht x B 3 B 1 B 2 U 1 U 2. It follows tht U 1 U 2 elongs to T y definition. Exmple 2.3. We onsider two different sis B nd B on set X = R Consider sis B of irulr regions for topology on R 2. B = {B(x 0, y 0, r) R 2 : (x 0, y 0, r) R 2 R + }, where B(x 0, y 0, r) = {(x, y) R 2 : (x x 0 ) 2 + (y y 0 ) 2 < r 2 }. Generted topology T(B) is given y T(B) = {U R 2 : x U, (x, y, r) suh tht x B(x, y, r) U}. 4

5 B 1 x B 2 B 3 x B 2 B 1 Figure 2: Consider sis B of irulr regions nd sis B of retngulr regions on R 2. Intersetion of two sis elements is nother sis element for B, ut not for B. 2. Consider sis B of retngulr regions for topology on R 2. B = {B (x 0, y 0, r) R 2 : (x 0, y 0, r) R 2 R + }, where B (x 0, y 0, r) = {(x, y) R 2 : mx{ x x 0, y y 0 } < r}. Generted topology T(B ) is given y T(B ) = {U R 2 : x U, (x, y, r) suh tht x B (x, y, r) U}. We show typil sis elements of B nd B in Figure 2. Oserve tht intersetion of two sis elements B 1, B 2 B is not sis element. But, y definition of sis, there exists n element B 3 B 1 B 2. On the other hnd, intersetion of two sis elements B 1, B 2 B is sis element itself. It n e shown tht T(B) = T(B ). Exmple 2.4. Let X e ny set, then olletion of ll singletons is sis for disrete topology on X. We will show olletion of ll singletons B = {{x} : x X} is sis. Covering whole set. Clerly X = x X = {x}. Bsis inside intersetion. Let x y, then {x} {y} =, so seond ondition is vuously true. 5

6 Lemm 2.5. Let X e set, nd B e sis for topology T on X. Then T equls the olletion of ll unions of elements of B. Proof. Let B = {B i : i I}. Sine B T nd T is topology, hene for ll J I, we hve j J B j T. Tht is, the olletion of union of elements in B is in T. Conversely, let U T e ny open suset of X. For ny x X, we n find B x B suh tht x B x U y definition of B. Hene, we hve U = x U B x. Tht is, ny element of T is union of sis elements in B. Remrk 3. There is no unique wy of representing n open set s union of sis elements. Thus, topologil sis is quite different to liner lger sis. 6