Movement to and from closed areas was tightly controlled. Foreigners were prohibited from entering them and local citizens were under stringent restrictions. They had to have special permission to travel there or leave, and anyone seeking residency was required to undergo vetting by the NKVD and its successor agencies. Access to some closed cities was physically enforced by surrounding them with barbed wire fences monitored by armed guards.
Closed cities were established in the Soviet Union from the late 1940s onwards under the euphemistic name of "post boxes", referring to the practice of addressing post to them via mail boxes in other cities. They fell into two distinct categories.
The locations of the first category of the closed cities were chosen for their geographical characteristics. They were often established in remote places situated deep in the Urals and Siberia, out of reach of enemy bombers. They were built close to rivers and lakes that were used to provide the large amounts of water needed for heavy industry and nuclear technology. Existing civilian settlements in the vicinity were often used as sources of construction labour. Although the closure of cities originated as a strictly temporary measure that was to be normalized under more favorable conditions, in practice the closed cities took on a life of their own and became a notable institutional feature of the Soviet system.
The level of most closed lakes is unstable because if runoff into the lake is lessened, the water balance of a closed lake is altered, and the amount of water in the lake falls. This is what has caused the shrinkage of the Aral Sea, formerly the world's second largest closed lake. Similarly, if runoff into a closed lake is increased, then the level will increase because evaporation is not likely to increase at all - let alone enough to stabilise the level of the lake.
In point set topology, a set A is closed if it contains all its boundary points.
A topological space X is disconnected if there exist disjoint, nonempty, closed subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.
The first use of regularly scheduled closed captioning on American television occurred on March 16, 1980. Sears had developed and sold the Telecaption adapter, a decoding unit that could be connected to a standard television set. The first programs seen with captioning were a Disney's Wonderful World presentation of the film Son of Flubber on NBC, an ABC Sunday Night Movie airing of Semi-Tough, and Masterpiece Theatre on PBS.
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.
Fluctuation in the level of closed lakes is therefore much more useful in paleoclimatology than are studies of open lakes which can reduce the level of outflow if inflow decreases.
An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of this characterization is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space X, because whether or not a sequence or net converges in X depends on what points are present in X.
In a closed lake (see endorheic drainage), no water flows out, and water which is not evaporated will remain in a closed lake indefinitely. This means that closed lakes are usually saline, though this salinity varies greatly from around three parts per thousand for most of the Caspian Sea to as much as 400 parts per thousand for the Dead Sea. Only the less salty closed lakes are able to sustain life, and it is completely different from that in rivers or freshwater open lakes. Closed lakes typically form in areas where evaporation is greater than rainfall, although most closed lakes actually obtain their water from a region with much higher precipitation than the area around the lake itself, which is often a depression of some sort.
In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.
Sets that can be constructed as the union of countably many closed sets are denoted F σ sets. These sets need not be closed.
The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.
Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection.
Infectious disease presents particular challenges to closed communities; external action (from the government or outside medical personnel) may assist in stopping the spread of the disease.
Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. Stone-Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.
A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than 2.
A Closed Area Permit is a document issued by the Hong Kong Police Force to allow for people with ties or residents in the area to travel in and out of the Frontier Closed Area. Visitors to the Mai Po Marshes are also required to apply for a Mai Po Marshes Entry Permit from the Agriculture, Fisheries and Conservation Department.
Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouville numbers, EL numbers and elementary numbers. The Liouville numbers, denoted L (not to be confused with Liouville numbers in the sense of rational approximation), form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in. L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in, denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "Exponential-Logarithmic" and as an abbreviation for "elementary".