Math 2.0

General topology Definition

Hemicontinuity

Builds on Topological spaceMultivalued functionOpen setNeighbourhoodContinuous map

Definition

Let XX and YY be topological spaces. Let Γ\Gamma be a multivalued function from XX to YY, that is, a map that sends each x∈Xx \in X to a nonempty subset Γ(x)⊆Y\Gamma(x) \subseteq Y. Let x0∈Xx_0 \in X.

  1. Γ\Gamma is upper hemicontinuous at x0x_0 if the following holds for every open set V⊆YV \subseteq Y with Γ(x0)⊆V\Gamma(x_0) \subseteq V. There is a neighbourhood UU of x0x_0 with Γ(x)⊆V\Gamma(x) \subseteq V for all x∈Ux \in U.
  2. Γ\Gamma is lower hemicontinuous at x0x_0 if the following holds for every open set V⊆YV \subseteq Y with Γ(x0)∩V≠∅\Gamma(x_0) \cap V \ne \emptyset. There is a neighbourhood UU of x0x_0 with Γ(x)∩V≠∅\Gamma(x) \cap V \ne \emptyset for all x∈Ux \in U.

Γ\Gamma is upper (lower) hemicontinuous if it is so at every point of XX, and continuous if it is both.

Remarks

Some authors say upper and lower semicontinuous instead of hemicontinuous. If Γ(x)={f(x)}\Gamma(x) = \set{f(x)} for a function f ⁣:X→Yf \colon X \to Y, then each of the two conditions is equivalent to the continuity of ff at x0x_0.

Example: let Γ ⁣:ℝ→ℝ\Gamma \colon \R \to \R with Γ(0)=[0,1]\Gamma(0) = [0, 1] and Γ(x)={0}\Gamma(x) = \set{0} for x≠0x \ne 0. It is upper hemicontinuous at 00: an open V⊇[0,1]V \supseteq [0, 1] contains 00, so Γ(x)⊆V\Gamma(x) \subseteq V for all xx.

Non-example: the same Γ\Gamma is not lower hemicontinuous at 00. The open set V=(1/2,2)V = (1/2, 2) meets Γ(0)\Gamma(0), but Γ(x)∩V=∅\Gamma(x) \cap V = \emptyset for every x≠0x \ne 0.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Continuous map, concept, no evidence levelContinuous mapconceptMultivalued function, concept, no evidence levelMultivalued functionconceptNeighbourhood, concept, no evidence levelNeighbourhoodconceptOpen set, concept, no evidence levelOpen setconceptTopological space, concept, no evidence levelTopological spaceconceptHemicontinuity, definition, no evidence levelHemicontinuitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Continuous map concept
  2. Multivalued function concept
  3. Neighbourhood concept
  4. Open set concept
  5. Topological space concept
  6. Hemicontinuity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hemicontinuity
About this page
TagCZCANZNKPJ
URIm2://CZCANZNKPJ
Kinddefinition
WritingW0 stub
MSC54C60, 26E25
DefinesHemicontinuity
UsesTopological space, Multivalued function, Open set, Neighbourhood, Continuous map
Changed2 November 2026

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