$\endgroup$ – … 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. 5.1. $\begingroup$ @user170039 - So, is it possible then to have a discrete topology on the set of all real numbers? That is, T discrete is the collection of all subsets of X. A set is discrete in a larger topological space if every point has a neighborhood such that . In nitude of Prime Numbers 6 5. I think not, but the proof escapes me. Topology of the Real Numbers In this chapter, we de ne some topological properties of the real numbers R and its subsets. We say that two sets are disjoint 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. Cite this chapter as: Holmgren R.A. (1994) The Topology of the Real Numbers. Continuous Functions 12 8.1. What makes this thing a continuum? Quotient Topology … The question is: is there a function f from R to R* whose initial topology on R is discrete? I mean--sure, the topology would have uncountably many subsets of the reals, but conceptually a discrete topology on the reals is possible, no? De ne T indiscrete:= f;;Xg. Subspace Topology 7 7. If anything is to be continuous, it's the real number line. Let Xbe any nonempty set. Product, Box, and Uniform Topologies 18 11. 52 3. discrete:= P(X). Therefore, the closure of $(a,b)$ is … Perhaps the most important infinite discrete group is the additive group ℤ of the integers (the infinite cyclic group). A Theorem of Volterra Vito 15 9. Universitext. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Then T discrete is called the discrete topology on X. Compact Spaces 21 12. Example 3.5. 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. Another example of an infinite discrete set is the set . For example, the set of integers is discrete on the real line. The intersection of the set of even integers and the set of prime integers is {2}, the set that contains the single number 2. 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 … Typically, a discrete set is either finite or countably infinite. 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. The points of are then said to be isolated (Krantz 1999, p. 63). Homeomorphisms 16 10. Then consider it as a topological space R* with the usual topology. Product Topology 6 6. The real number line [math]\mathbf R[/math] is the archetype of a continuum. 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