discrete:= P(X). Continuous Functions 12 8.1. 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. Typically, a discrete set is either finite or countably infinite. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. In: A First Course in Discrete Dynamical Systems. Compact Spaces 21 12. If $\tau$ is the discrete topology on the real numbers, find the closure of $(a,b)$ Here is the solution from the back of my book: Since the discrete topology contains all subsets of $\Bbb{R}$, every subset of $\Bbb{R}$ is both open and closed. For example, the set of integers is discrete on the real line. $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? Let Xbe any nonempty set. That is, T discrete is the collection of all subsets of X. Homeomorphisms 16 10. The question is: is there a function f from R to R* whose initial topology on R is discrete? Consider the real numbers R first as just a set with no structure. Example 3.5. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. A Theorem of Volterra Vito 15 9. Quotient Topology 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? Then T discrete is called the discrete topology on X. Then consider it as a topological space R* with the usual topology. A set is discrete in a larger topological space if every point has a neighborhood such that . In nitude of Prime Numbers 6 5. The real number line $\mathbf R$ is the archetype of a continuum. If anything is to be continuous, it's the real number line. TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by ââ©.â Aâ© B is the set of elements which belong to both sets A and B. Product Topology 6 6. 52 3. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. Subspace Topology 7 7. $\endgroup$ â â¦ We say that two sets are disjoint Perhaps the most important infinite discrete group is the additive group â¤ of the integers (the infinite cyclic group). The points of are then said to be isolated (Krantz 1999, p. 63). Another example of an infinite discrete set is the set . Product, Box, and Uniform Topologies 18 11. What makes this thing a continuum? 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. Universitext. De ne T indiscrete:= f;;Xg. Open sets Open sets are among the most important subsets of R. 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