Consider the real numbers R first as just a set with no structure. Product Topology 6 6. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. For example, the set of integers is discrete on the real line. De ne T indiscrete:= f;;Xg. discrete:= P(X). $\endgroup$ – … The points of are then said to be isolated (Krantz 1999, p. 63). Another example of an infinite discrete set is the set . Homeomorphisms 16 10. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Compact Spaces 21 12. Subspace Topology 7 7. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. 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. Let Xbe any nonempty set. Product, Box, and Uniform Topologies 18 11. We say that two sets are disjoint Typically, a discrete set is either finite or countably infinite. 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. Then T discrete is called the discrete topology on X. In: A First Course in Discrete Dynamical Systems. Example 3.5. The question is: is there a function f from R to R* whose initial topology on R is discrete? That is, T discrete is the collection of all subsets of X. Therefore, the closure of $(a,b)$ is … A set is discrete in a larger topological space if every point has a neighborhood such that . 52 3. Then consider it as a topological space R* with the usual topology. The real number field ℝ, with its usual topology and the operation of addition, forms a second-countable connected locally compact group called the additive group of the reals. Continuous Functions 12 8.1. $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? In nitude of Prime Numbers 6 5. 5.1. What makes this thing a continuum? I think not, but the proof escapes me. Perhaps the most important infinite discrete group is the additive group ℤ of the integers (the infinite cyclic group). Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. Open sets Open sets are among the most important subsets of R. 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