The real number line [math]\mathbf R[/math] is the archetype of a continuum. 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. Subspace Topology 7 7. 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. 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. A set is discrete in a larger topological space if every point has a neighborhood such that . $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? Product, Box, and Uniform Topologies 18 11. In: A First Course in Discrete Dynamical Systems. Another example of an infinite discrete set is the set . discrete:= P(X). Typically, a discrete set is either finite or countably infinite. 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 … Consider the real numbers R first as just a set with no structure. Then T indiscrete is called the indiscrete topology on X, or sometimes the trivial topology on X. Quotient Topology … TOPOLOGY AND THE REAL NUMBER LINE Intersections of sets are indicated by “∩.” A∩ B is the set of elements which belong to both sets A and B. De ne T indiscrete:= f;;Xg. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. Example 3.5. We say that two sets are disjoint 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. $\endgroup$ – … What makes this thing a continuum? I think not, but the proof escapes me. 5.1. The points of are then said to be isolated (Krantz 1999, p. 63). 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. Homeomorphisms 16 10. Compact Spaces 21 12. Therefore, the closure of $(a,b)$ is … Let Xbe any nonempty set. Then consider it as a topological space R* with the usual topology. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. For example, the set of integers is discrete on the real line. 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