Sets, Infinity, and Mappings
نویسنده
چکیده
These notes discuss sets, mappings between sets, and the notion of countably infinite. This material supplements the lecture on countably and uncountably infinite sets in the EE 503 probability class. The material is useful because probability theory is defined over abstract sets, probabilities are defined as measures on subsets, and random variables are defined by mappings from an abstract set to a real number. Further, it is important to understand the difference between “countably infinite” and “uncountably infinite.” One reason is that probability techniques for uncountably infinite sets are different than those for countably infinite sets. The final sections of the notes discuss related material that is not part of the course but may be of interest. This includes Cantor’s theorem for power sets and famous paradoxes. I. INFINITY IN A NUTSHELL
منابع مشابه
$r$-fuzzy regular semi open sets in smooth topological spaces
In this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. By using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. Also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Moreover, we investigate the ...
متن کاملFixed point theorems for $alpha$-$psi$-contractive mappings in partially ordered sets and application to ordinary differential equations
In this paper, we introduce $alpha$-$psi$-contractive mapping in partially ordered sets and construct fixed point theorems to solve a first-order ordinary differential equation by existence of its lower solution.
متن کاملGeneralized Regular Fuzzy Irresolute Mappings and Their Applications
In this paper, the notion of generalized regular fuzzy irresolute, generalized regular fuzzy irresolute open and generalized regular fuzzy irresolute closed maps in fuzzy topological spaces are introduced and studied. Moreover, some separation axioms and $r$-GRF-separated sets are established. Also, the relations between generalized regular fuzzy continuous maps and generalized regular fuzzy ...
متن کاملFuzzy $e$-regular spaces and strongly $e$-irresolute mappings
The aim of this paper is to introduce fuzzy ($e$, almost) $e^{*}$-regular spaces in $check{S}$ostak's fuzzy topological spaces. Using the $r$-fuzzy $e$-closed sets, we define $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points and their properties. Moreover, we investigate the relations among $r$-($r$-$theta$-, $r$-$etheta$-) $e$-cluster points, $r$-fuzzy ($e$, almost) $e^{*}$-regular spaces a...
متن کاملNonexpansive Mappings Defined on Unbounded Domains
In the study of nonexpansive mappings and fixed point theory the domain of the mapping is usually assumed to be bounded or, as in certain approximation results (see, e.g., [20]), fixed points are assumed to exist. However in [21] Penot used uniform asymptotic concepts which he had earlier introduced in [22] to extend the Browder-Göhde-Kirk theorem to unbounded sets. The term “asymptotic” is use...
متن کامل