The usual topology on such a state spaces can be given by the metric ρ which assigns to two sequences S = (s i) and T = (t i) a distance 2 − k if k is the smallest absolute value of an index i for which the corresponding elements s i and t i are different. For example, recall that we described the usual topology on R explicitly as follows: T usual = fU R : 8x2U;9 >0 such that (x ;x+ ) Ug; We then remarked that the open sets in this topology are precisely the familiar open intervals, along with their unions. Recall: pAXBqA AAYBAand pAYBqA AAXBA A set C is a closed set if and only if it contains all of its limit points. Let with . First examples. Example 12. But is not -regular. We will now look at some more examples of bases for topologies. For example, the following topology (the trivial topology) is a perfectly fine topology for $\mathbb R$: $$ \{\varnothing,\mathbb R\}. Then is a -preopen set in as . Example 5. Thus we have three different topologies on R, the usual topology, the discrete topol-ogy, and the trivial topology. Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. In the de nition of a A= ˙: Example 11. Hausdorff or T2 - spaces. Here are two more, the first with fewer open sets than the usual topology… If we let O consist of just X itself and ∅, this defines a topology, the trivial topology. Example 1. 2Provide the details. In Example 9 mentioned above, it is clear that is a -open set; thus it is --open, -preopen, and --open. We also know that a topology … (a, b) = (a, ) (- , b).The open intervals form a base for the usual topology on R and the collection of all of these infinite open intervals is a subbase for the usual topology on R.. Example 6. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. (Finite complement topology) Define Tto be the collection of all subsets U of X such that X U either is finite or is all of X. Example 2.1.8. See Exercise 2. Corollary 9.3 Let f:R 1→R1 be any function where R =(−∞,∞)with the usual topology (see Example 4), that is, the open sets are open intervals (a,b)and their arbitrary unions. 94 5. (Usual topology) Let R be a real number. Example: If we let T contain all the sets which, in a calculus sense, we call open - We have \R with the standard [or usual] topology." Example 1.3.4. Let be the set of all real numbers with its usual topology . Then in R1, fis continuous in the −δsense if and only if fis continuous in the topological sense. Example: [Example 3, Page 77 in the text] Xis a set. Thus -regular sets are independent of -preopen sets. The following theorem and examples will give us a useful way to define closed sets, and will also prove to be very helpful when proving that sets are open as well. Example 1.2. Example 1, 2, 3 on page 76,77 of [Mun] Example 1.3. $$ (You should verify that it satisfies the axioms for a topology.) Definition 1.3.3. Let X be a set. Then V={ GR: Vx EG 38>0 such that (*-8,x+8)¢GUR, is the usual topology on R. 6.1. Every open interval (a, b) in the real line R is the intersection of two infinite open intervals (a, ) and (- , b) i.e. Definition 6.1.1. Interior and isolated points of a set belong to the set, whereas boundary and accumulation points may or may not belong to the set. T f contains all sets whose complements is either Xor nite OR contains ˜ and all sets whose complement is nite. (Discrete topology) The topology defined by T:= P(X) is called the discrete topology on X. Topology of the Real Numbers When the set Ais understood from the context, we refer, for example, to an \interior point." But is not -regular because . topology.