the closed sets as the fixed points of an operator on the power set of A net is a generalisation of the concept of sequence. A topology is completely determined if for every net in the set of its accumulation points is specified. Comparison of topologies A variety of topologies can be placed on a set to form a topological space. When every set in a topology is also in a topology and is a subset of we say that is than and is than A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms and are sometimes used in place of finer and coarser, respectively. The terms and are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. The collection of all topologies on a given fixed set forms a complete lattice: if is a collection of topologies on then the meet of is the intersection of and the join of is the meet of the collection of all topologies on that contain every member of Continuous functions A function between topological spaces is called continuous if for every and every neighbourhood of there is a neighbourhood of such that This relates easily to the usual definition in analysis. Equivalently, is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In category theory, one of the fundamental categories is Top, which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory. Examples of topological spaces A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete