In mathematics, the term dense has at least three different meanings.
A subsetA of a topological spaceX is said to be dense if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X. Equivalently, every nonempty open subset of X intersects A, or in other words: the interior of the complement of A is empty. As an example, the set of rational numbers is a dense subset of the real numbers.
A partial order on a setS is said to be dense if, for all x and y in S for which x < y, there is a z in S such that x < z < y. The rational numbers with the ordinary ordering are a densely ordered set in this sense, as are the real numbers.
A subset B of a partially ordered set A is dense in A if for any x < y in A, there is some z in B such that x < z < y. In case the order is a linear order, then B is dense in A in the present sense if and only if B is dense in the order topology on A. Hence the first two meanings above are related.
Note that the first notion of "dense" depends on the surrounding space, while the second notion is completely internal to the ordered set. The rationals in [0,1] for instance are dense as an ordered set and they are dense in the space [0,1] but they are not dense in the real numbers.
A densetopology may induce high interference, which, in turn, reduces the effective network capacity due to limited spatial reuse and may cause unnecessarily high energy consumption.
Topology control for ad hoc networks aims to maintain a specified topology, such as requiring that the network be connected.
The desired effect of topology control is to reduce energy consumption, reduce MAC layer interference between adjacent nodes, and to increase the effective network capacity.
Eg in cofinite topology, since only finite sets are closed, either X, if X is finite, or any countably infinite set is dense hence X is separable.
In real life you only ever need the standard metric topology or some other topology of continuity of functions that you pick a posteri to make things work (eg topology of pointwise convergence, topology of uniform convergence), or the zariski topology, or the compact open topology.
By example show that the zariski topology on R^2 is not the product topology from the zariski topology on two copies of R. It's not hard but is an (the only?) interesting example of a case where the product topology is not what you think it might be.
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