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This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets. Closed function: maps closed sets to closed sets.
[1] [2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph. [3] This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily ...
The usual proof of the closed graph theorem employs the open mapping theorem.It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see closed graph theorem § Relation to the open mapping theorem (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)
Closed graph theorem for set-valued functions [6] — For a Hausdorff compact range space , a set-valued function : has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all .
A closed rectangle does not have a neighbourhood on any of its corners or its boundary since there is no open set containing any corner. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
The presence of reactants in an open beaker is an example of an open system. Here the boundary is an imaginary surface enclosing the beaker and reactants. It is named closed , if borders are impenetrable for substance, but allow transit of energy in the form of heat, and isolated , if there is no exchange of heat and substances.
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that () is the intersection of the closed sets containing X. This equivalence remains true for partially ordered sets with the greatest-lower-bound property , if one replace "closed sets" by "closed elements" and ...