A Course in Point Set Topology (Undergraduate Texts in Mathematics)

A) allow , and for every n in enable . provide the ensuing product area the field topology and outline the functionality by means of . convey that f isn't really non-stop, even if is constant for every n. [Hint: research . ] (b) back allow , and for every n in permit X n  = { 0, 1}. convey that the field topology on X is the discrete topology. for this reason X isn't compact, even supposing each one coordinate area is compact. (4)Find an instance of a set of hooked up topological areas such that the product house with the field topology isn't really attached. (5)Let be a suite of topological areas, and permit have the product topology. exhibit that X is separable if and provided that I is countable and every X i is separable. (6)Let be a suite of topological areas, and allow X = ∏ i X i've got the product topology. If, for every i in I, C i is part of X i , is C = ∏ i ∈ I C i an element of X? (7)Let be a suite of topological areas, and permit X = ∏ i X i've got the product topology. (a) express that if X is in the community hooked up, then every one X i is in the neighborhood hooked up. (Hint: Proposition 2. four. 19 can be beneficial. ) (b) express that the communicate is fake through giving a counterexample. (See Exercise 2. four. 7. ) (c) in the event you suppose that every X i is in the neighborhood attached, are you able to provide an extra speculation that means that X is in the neighborhood attached? (8)Explore the questions raised within the previous workout for neighborhood pathwise connectedness. 2. 7 Nets Now we generalize the suggestion of a chain with an idea that works virtually besides for an arbitrary topological house as sequences paintings for metric areas. references the place the reader will locate all this fabric and extra are [6] and [4]. remember the definition of ordered set given in � A. four. Definition 2. 7. 1. A directed set is ordered set I with the valuables that if i, j ∈ I, then there's a okay in I with i, j ≤ k. Example 2. 7. 2. (a) is a directed set. (b)The set of all integers is a directed set. (c)Let E be any set and order 2 E via inclusion. that's, if A, B ∈ 2 E , then A ≤ B ability . It follows that 2 E is a directed set. (d)Again, if E is any set, is the gathering of all finite subsets of E, and is ordered via inclusion, then is a directed set. (e)Let X be a topological house, and for a few x zero in X enable denote the gathering of all neighborhoods of x zero. you'll be able to see that could be a directed set below opposite inclusion; that's, if , claim U ≥ V to intend that . allow us to point out anything that's rapid from the definition of a directed set and that might be used frequently: if I is directed and , then there's an i in I with . (Verify! ) Definition 2. 7. three. A web in a collection X is a couple (x, I), the place I is a directed set and x is a functionality from I into X. we'll frequently (usually) write the internet as {x i : i ∈ I} or {x i } if the directed set I is known; the functionality notation {x(i): i ∈ I} will seldom be used to indicate a web. word that each series is a internet. Example 2. 7. four. (a)Let 2 E be the directed set from Example 2. 7. 2(c), and for every A ∈ 2 E decide some degree x A in A. Then {x A } is a web. (b)Again permit E be a collection, and enable be the gathering of all nonempty finite subsets of as in Example 2.