Topological Space
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High Quality Content by WIKIPEDIA articles! Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. A variety of topologies can be placed on a set to form a topological space. When every set in a topology 1 is also in a topology 2, we say that 2 is finer than 1, and 1 is coarser than 2. A proof which relies only on the...
High Quality Content by WIKIPEDIA articles! Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. A variety of topologies can be placed on a set to form a topological space. When every set in a topology 1 is also in a topology 2, we say that 2 is finer than 1, and 1 is coarser than 2. A proof which 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 larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker 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.
