Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20140114 19:54:24
Examples of Open and Close SetsIf is a collection of subsets of such that:(1) the union of any collection of sets that are elements of belongs to ; (2) the intersection of any finite collection of sets that are elements of belong to ; (3)the empty set and belong to . Then, elements of are open sets of the topological space . A set which is a subset of X is closed in the space if its complement is open (i.e., X \ F ∈ Ω). Could someone give me examples of a) Closed Sets b) Open Sets c) sets which are both open and closed; d) sets which are neither closed nor open. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #2 20140114 20:47:13
Re: Examples of Open and Close SetsFirst you need an example of a topological space. Let and . Then (a) (its complement in is ) (b) (or any member of ) (c) and (these are always both open and closed in any topological space; they are called clopen sets) (d) 134 books currently added on Goodreads #3 20140114 21:09:19
Re: Examples of Open and Close SetsHi; 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #4 20140114 21:32:42
Re: Examples of Open and Close Sets
In , any closed interval is a closed set. This includes singleton sets as .
In , an example of a closed set would be the interior and boundary of a circle (or square, or polygon, or any simply connected plane figure). 134 books currently added on Goodreads #5 20140115 01:39:02
Re: Examples of Open and Close SetsWhy is it a closed set? What is the compliment of a closed interval? When on R, what exactly is the topological space? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #6 20140115 05:49:53
Re: Examples of Open and Close SetsThe open sets in are arbitrary unions of open intervals. The complement of is , a union of two open intervals; therefore it’s open (so is closed). The open sets in are more complicated to describe. Basically think of an “open figure” as a connected region of the complex plane that does not include the boundary. For example, the circle , which does not include points on the circumference. (Note that such a region need not be bounded; e.g. the half plane is an “open figure”.) Then the open sets in are arbitrary unions of “open figures”. All this may be very confusing for you, I know, but this is what and are as topological spaces. It would be much simpler to treat them as metric spaces instead. Last edited by Nehushtan (20140115 06:03:41) 134 books currently added on Goodreads #7 20140116 04:13:20
Re: Examples of Open and Close SetsIs it possible for other clopen sets to exist than phi and x? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda 