Example 1.2. We also know that a topology … Example 1.3.4. 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. 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. A set C is a closed set if and only if it contains all of its limit points. Then is a -preopen set in as . In the de nition of a A= ˙: Example 2.1.8. Here are two more, the first with fewer open sets than the usual topology… Example: [Example 3, Page 77 in the text] Xis a set. 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, the following topology (the trivial topology) is a perfectly fine topology for $\mathbb R$: $$ \{\varnothing,\mathbb R\}. We will now look at some more examples of bases for topologies. Example 1, 2, 3 on page 76,77 of [Mun] Example 1.3. See Exercise 2. In Example 9 mentioned above, it is clear that is a -open set; thus it is --open, -preopen, and --open. First examples. Let be the set of all real numbers with its usual topology . Thus we have three different topologies on R, the usual topology, the discrete topol-ogy, and the trivial topology. 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." Let X be a set. Example 1. (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.. Recall: pAXBqA AAYBAand pAYBqA AAXBA topology. Definition 1.3.3. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. (Discrete topology) The topology defined by T:= P(X) is called the discrete topology on X. Example 12. 94 5. Definition 6.1.1. Then V={ GR: Vx EG 38>0 such that (*-8,x+8)¢GUR, is the usual topology on R. 6.1. Example 6. Then in R1, fis continuous in the −δsense if and only if fis continuous in the topological sense. But is not -regular because . 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. Thus -regular sets are independent of -preopen sets. 2Provide the details. If we let O consist of just X itself and ∅, this defines a topology, the trivial topology. T f contains all sets whose complements is either Xor nite OR contains ˜ and all sets whose complement is nite. Hausdorff or T2 - spaces. Interior and isolated points of a set belong to the set, whereas boundary and accumulation points may or may not belong to the set. Topology of the Real Numbers When the set Ais understood from the context, we refer, for example, to an \interior point." $$ (You should verify that it satisfies the axioms for a topology.) But is not -regular. Example 5. (Usual topology) Let R be a real number. Example 11. Let with . (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. 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