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Some Classical Results
Published in James K. Peterson, Basic Analysis III, 2020
We first need to return to the notion of nowhere dense sets we introduced earlier in Chapter 5. Recall a subset of a normed linear space is nowhere dense if its closure has empty interior. This is defined in Definition 5.4.3. In (Peterson (18) 2020), a standard project is to construct Cantor sets and we prove a Cantor set cannot contain any interval. Hence it is a nowhere dense subset of [0,1]. We also prove that compact subsets of infinite dimensional normed linear spaces are nowhere dense in Theorem 5.4.5. We now want to study these sets more carefully.
Banach–Steinhaus Theorem
Published in P.N. Natarajan, Functional Analysis and Summability, 2020
Since A¯ is closed, A is nowhere dense in X if and only if A¯ is nowhere dense in X. A subset of a nowhere dense set is itself nowhere dense.
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The Baire Categories. It is convenient at this point to mention a special property of complete metric spaces. A subspace A of a topological space X is said to be nowhere dense in X if the interior of its closure is empty: intA¯=∅ $ \mathrm{int} \overline{A} = \emptyset $ . For example, the integers Z $ \mathcal Z $ are nowhere dense in R $ \mathbb R $ . A topological space X is said to be of the first category if X is the countable union of nowhere dense subsets of X. Otherwise, X is of the second category. These are called the Baire categories. For example, the rationals Q $ \mathbb Q $ are of the first category, because the singleton sets {q} $ \{q\} $ are nowhere dense in Q $ \mathbb Q $ , and Q $ \mathbb Q $ is the countable union of such sets. The real line R $ \mathbb R $ is of the second category. Indeed, that every complete metric space is of the second category is the premise of the following basic theorem.
Smoothing method in multi-criteria transportation network equilibrium problem
Published in Optimization, 2019
We recall that a nonempty set is said to be nowhere dense in K if its closure has empty interior in K. Equivalently, A is nowhere dense if its complement is a set with dense interior in K. For denotes the O/D pair connected by the path and denotes the set of all paths connecting this O/D pair. For a feasible flow y and an O/D pair , is the set of all cost functions ; , called the set of minimal elements of , consists of vectors such that there is no satisfying . In view of the proof of Theorem 4.1 of [15], given any open set U in K, there is an open subset and an index set such that Let denote the Euclidean distance from the point to the set in . Then for every and one has The following notations will also be used: is the set of all indices j such that satisfying .