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Then f : X → Y is continuous but its graph is not closed in X × Y. [4] If X is any space then the identity map Id : X → X is continuous but its graph, which is the diagonal Gr Id := { (x, x) : x ∈ X }, is closed in X × X if and only if X is Hausdorff. [7] In particular, if X is not Hausdorff then Id : X → X is continuous but not closed.
Closed graph theorem [5] — If : is a map from a topological space into a Hausdorff space, then the graph of is closed if : is continuous. The converse is true when Y {\displaystyle Y} is compact .
The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. [10] Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space ...
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone. The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ {\displaystyle \delta } depends on ε ...
Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function () =, which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point.
A non-closed curve may also be called an open curve. If the domain of a topological curve is a closed and bounded interval = [,], the curve is called a path, also known as topological arc (or just arc). A curve is simple if it is the image of an interval or a circle by an injective continuous function.
called the nth homology group of X. The elements of H n (X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous. [6] A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map.