22M 132 Fall 07 J Simon Some highlights on Countability and Separation Properties Relates to text Sec 30 36 Introduction There are two basic themes to the next several sections a What properties of a topology allow us to conclude that the topology is given by a metric b What properties of a space allow us to conclude that the space actually is homeomorphic to a subspace of Rn or at least a subspace of R Countability Properties Here are several properties of spaces all saying that the topology or some key feature of it can be described in terms of countably many pieces of information The names are historical they are not very descriptive or otherwise useful but you should know them since they are used in the literature 1 First axiom of countability The space X T is called first countable if the topology has a countable local basis at each point x X 2 Second axiom of countability The space X T is called second countable if the topology T has a countable basis 3 Separable The space X T is called separable if X contains a countable dense subset Recall a subset A X is called dense in X if the closure A is all of X i e each open set contains at least one point of A 4 Lindelo f property The space X T is called a Lindelo f space if each open cover of X has a countable sub cover The familiar space Rn with the standard topology has all of the above properties proof below For more general spaces we can ask many questions Do any of these properties imply others If a space X has one of the properties do all subspaces of X have the property In that always happens we would call the property hereditary If we have a family of spaces with one of these properties does the cartesian product have the property If f X Y is a continuous surjection and X has one of the properties must Y also have the property If X T has one of these properties and T is a coarser resp finer topology must X T have the property We will focus on just some highlights c J Simon all rights reserved page 1 Theorem text 30 3 Countable basis all the other countability properties Proof Suppose B is a countable basis for the topology on X a Countable local basis Let x X and let U be any neighborhood of x Since B is a basis for the topology U is a union of elements of B Thus there exists an element B B such that x B U So the set B is a countable local basis for each point x X b Separable For each nonempty set B B pick a point xB B Since B is countable the set xB B B is countable Since each open set is a union of elements of B each nonempty open set U contains at least one of the sets B and so xB U Thus xB B B is dense in X c Lindelo f Let U J be an open cover of X We want to prove there exists a countable subcover by somehow using the existence of a countable basis B for the topology For convenience to make the exact argument a little simpler assume that one of the sets U is actually the empty set Or adjoin one additional set U0 to the covering The idea of the proof is to use the elements of B to point to certain special U s Specifically for each set B B we will select one set UB from among the U s as follows First ask if there exists at least one of the open sets U containing that set B If not let UB U0 If the set B is contained in some U then pick one such U and call it UB We might pick the same U corresponding to several B s because a given U usually contains many basis sets but we have at most one U chosen for each B so the set UB B B is countable We now show that UB B B covers X Let x X We shall prove that at least one of the sets UB contains x Since the U s cover X there is some U containing x Since B is a basis there exists B B with x B U Since that basis set B is contained in some U B is one of the basis sets for which we chose a set UB B So x B UB in particular x UB The previous theorem says that having a countable basis for the topology is the strongest of the countability properties The next example shows that it is strictly stronger that is the other properties do not imply it Example ex 3 page 192 The space R is first countable separable and Lindelo f but not second countable Proof The details are given in the text you should be able to prove R is first countable and separable and that it is not second countable You are not required to know the proof that R is Lindelo f The previous example shows some of the independence of the properties However in metric spaces the first countable separable and second countable properties are equivalent c J Simon all rights reserved page 2 Theorem Exercise 5 page 194 Suppose X is a metric space Then i X has countable local bases at each point ii X separable X has a countable basis iii X Lindelo f X has a countable basis Proof i The idea of countable local basis is precisely a generalization of the balls of radius 1 n in metric spaces The set B x n1 is a local basis at x ii Let xn n N be a countable dense set in X For each xn let Bn be the set of all open balls centered at xn with rational radius Then the set Bn is countable for each n so the set B Bn n N is countable We claim this set B is a basis The proof is an exercise in using the triangle inequality that the metric satisfies You can work out the details here is the idea Take any open set U X We want to show that U is a union of our alleged basis elements Let y be any point of U we shall show that there is one of our alleged basis sets B such that y B U The point y is contained in some ball inside U Now look at an 100 ball around y This ball must contain some point xn from our countable dense subset So the distance from xn to y is less than 100 Then by picking a rational radius slightly larger than 100 we can find a rational radius ball centered at xn containing y and contained in U iii For each n consider the open covering of X consisting of all balls of radius 1 n The …