Another example of an infinite discrete set is the set . discrete:= P(X). What makes this thing a continuum? Example 3.5. 5.1. Perhaps the most important infinite discrete group is the additive group â¤ of the integers (the infinite cyclic group). A Theorem of Volterra Vito 15 9. For example, the set of integers is discrete on the real line. TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by ââ©.â Aâ© B is the set of elements which belong to both sets A and B. $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Open sets Open sets are among the most important subsets of R. A collection of open sets is called a topology, and any property (such as â¦ I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? Universitext. If anything is to be continuous, it's the real number line. Continuous Functions 12 8.1. The real number line [math]\mathbf R[/math] is the archetype of a continuum. Therefore, the closure of $(a,b)$ is â¦ In: A First Course in Discrete Dynamical Systems. The points of are then said to be isolated (Krantz 1999, p. 63). A set is discrete in a larger topological space if every point has a neighborhood such that . Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. Product Topology 6 6. In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology.For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. Homeomorphisms 16 10. Consider the real numbers R first as just a set with no structure. That is, T discrete is the collection of all subsets of X. 52 3. Typically, a discrete set is either finite or countably infinite. Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. I think not, but the proof escapes me. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. The real number field â, with its usual topology and the operation of addition, forms a second-countable connected locally compact group called the additive group of the reals. In nitude of Prime Numbers 6 5. Subspace Topology 7 7. $\endgroup$ â â¦ Compact Spaces 21 12. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. Quotient Topology â¦ Let Xbe any nonempty set. We say that two sets are disjoint The question is: is there a function f from R to R* whose initial topology on R is discrete? De ne T indiscrete:= f;;Xg. 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