Accessibility

This page is incomprehensible to all but those who are up on there set theory notation. There is no reason for this, uhc and lhc are not that advanced of concepts. I'd add the appropriate clarity tag, but I forgot what it is. Pdbailey (talk) 03:47, 14 January 2008 (UTC)[reply]

I think the page also needs a figure, and an explanation of why these definitions are useful. Pdbailey (talk) 05:54, 14 January 2008 (UTC)[reply]

Introduction

What does it mean "the graph of the correspondence is closed from the left and from the right" and "the graph has no closed edges"? Either explain it better, or delete it: now, what should be the "intutitive" explanation is way more difficult that the actual definition. Manta (talk) 16:44, 27 November 2008 (UTC)[reply]

Questions from Patrickassonken

I don t understand in what sense the concept of continuity for single valued function is not extengible to multi valued functions. I think that as long as both spaces are endowed with a topology,we can extend the concept of continuity for single valued functions without issues. Also(my first langage is not english ,so I might be sorely wrong here) in the paragraph intitled "implications to continuity" ,the first sentence sounds like the definition of continuity is given by both hemicontinuities,which is not false but it is more accurate to refer to the link between continuity and hemicontinuity as a property.Patrickassonken 10:59pm ,july 4th 2008(ET) —Preceding unsigned comment added by Patrickassonken (talk • contribs) 03:00, 5 July 2008 (UTC)[reply]

Would you explain it better? Please give your definition of continuity for applications. Manta (talk) 16:44, 27 November 2008 (UTC)[reply]

It would seem to me that, using the topological idea of continuity, the natural extension to correspondences would be UHC (an open set in the range implies an open set in the domain, the image of which is contained in the other). However, if you use the sequential notion of continuity than UHC alone leaves something to be desired, namely that there may be points in the graph that cannot be approximated from the left or the right. Taking a quick look at Fudenburg and Tirole's "Game Theory" I see they remark that the Nash Equilibria (NE) of some game are not LHC (that is, as you vary the parameters). This suggests that, in the limit, there may be more equilibria available than there were for every non-limit value of the parameter. I suppose it would be desirable then to find games that yield NE correspondences which are both UHC and LHC, and then your approximations from the left or right would always have very meaningful content. I'm no game theorist, but this is my impression. Thedudeoflife (talk) 01:00, 26 October 2009 (UTC)[reply]

Why not semicontinuous

From the article:

One should be warned that the term upper semicontinuous, instead of upper hemicontinuous, has been more widespread in the literature. Then upper hemicontinuous is reserved for upper semicontinuity with respect to the weak topology.

So if semicontinuous is the more common terminology, why doesn't this article use that? It doesn't even avoid a name clash with semi-continuity according to this quote. Quietbritishjim (talk) 01:08, 9 August 2009 (UTC)[reply]

Which literature? Royden's textbook and the wiki page on semicontinuity only define the term semicontinuity for functions, not for correspondences. Thus, in my experience, the two terms have quite independent meanings and it's not obvious how semicontinuity of functions could generalize to correspondences and deliver the same ideas as captured in hemicontinuity. Perhaps someone could expand a little on that. Thedudeoflife (talk) 00:40, 26 October 2009 (UTC)[reply]

In his monograph "Optima and Equilibria" (Springer-Verlag, Berlin, 1993, chapter 9) Jean-Pierre Aubin distinguishes between (upper) hemi-continuous set-valued maps and (upper) semi-continuous set-valued maps. In proposition 9.1 (ibd.) he proves that any upper semi-continuous mapping is upper hemi-continuous. Yet one must not lose sight of the fact that Aubin presupposes Hilbert spaces. It is not clear to me whether hemi-continuity and semi-continuity are in general equivalent notions.Schojoha (talk) 17:25, 29 August 2011 (UTC)[reply]

I agree with Quietbritishjim (talk · contribs) in that semicontinuity is a much more common term than hemicontinuity. There may be some technical difference, but it's clear that terms are similar and in many settings they agree.

In Optima And Equillibria, Aubin has the following definitions (p. 137):

Definition 9.1. We shall say that a set-valued map $C$ is upper hemicontinuous at $x_{0}\in K$ if and only if for all $p\in Y^{*}$ , the function $x\to \sigma (C(x),p)$ is upper semi-continuous at $x_{0}$ . It is upper hemi-continuous if it is upper hemicontinuous at all points $x_{0}\in K$ .

Note that Aubin defines

x\mapsto \sigma (C(x),p)

to be the support function of

C

:

We use the fact that the images $C(x)$ are convex closed sets to represent them by their support functions $\forall p\in Y^{*}{\text{, we set }}\sigma (C(x),p)=\sup _{y\in C(x)}\langle p,y\rangle$

Definition 9.2. We shall say that a set-valued map $C$ from $K$ to $Y$ is upper semi-continuous at $x_{0}$ if, for all $\varepsilon >0$ , there exists a neighbourhood $N\left(x_{0}\right)$ of $x_{0}$ such that $C(x)\subset C\left(x_{0}\right)+\varepsilon B$ for all $x\in N\left(x_{0}\right)$ . It is upper semi-continuous if it is upper semi-continuous at all points $x_{0}\in K$ .

Definition 9.4. We shall say that a set-valued map $C$ from $K$ to $Y$ is lower semi-continuous at $x_{0}\in K$ if for any sequence $x_{n}$ converging to $x_{0}\in K$ , for all $y_{0}\in C\left(x_{0}\right)$ , there exists a sequence of elements $y_{n}\in C\left(x_{n}\right)$ converging to $y_{0}$ .

Definition 9.5. We shall say that a set-valued map $C$ is continuous (at $x_{0}$ ) if it is both lower and upper semi-continuous (at $x_{0}$ ).

Interestingly, he does not define "lower hemi-continuous". The-erinaceous-one (talk) 07:52, 1 August 2024 (UTC)[reply]

Sequential definitions

It would be useful to have the alternative definitions based on sequences, or equivalence theorems. —Preceding unsigned comment added by 72.33.43.134 (talk) 20:48, 7 June 2010 (UTC)[reply]

I think the sequential characterization is not valid in general topological spaces. It would appear to me that that the author presupposes metric spaces; cf. the textbook of Aubin and Frankowska ("Set-Valued Analysis", Birkhauser, Basel, 1990) Schojoha (talk) 17:38, 18 August 2011 (UTC)[reply]

I think the sequential characterization is for closed graph property, not for uhc. — Preceding unsigned comment added by Owichert2 (talk • contribs) 19:38, 23 September 2014 (UTC)[reply]

I added an assummption, "For a correspondence Γ : A → B with closed values," for the sequential definition. Otherwise, the function &Gamma(x) = (0,1) is a conterexample. A proof is in "Fixed-point theorem with application to economics and game theory", chapter 11, Proposition 11.11 — Preceding unsigned comment added by 194.199.27.238 (talk) 15:34, 5 November 2015 (UTC)[reply]

The sequential definition is for closed graph theorem. The added assumption required is that the range space is compact. Otherwise, we need to use the sub-sequential characterisation. (See Aliprantis and Border. 2005, "Infinite Dimensional Analysis", theorems 17.11 and 17.20. — Preceding unsigned comment added by Shanker akshay (talk • contribs) 02:10, 20 June 2017 (UTC)[reply]

Moreover, $A$ and $B$ need to be metric spaces. Se — Preceding unsigned comment added by Shanker akshay (talk • contribs) 02:18, 20 June 2017 (UTC)[reply]

This question

$\Gamma \left(x\right)=\left\{y\in R,\;\max \left(\left|x\right|,\left|y\right|\right)\leq 1\right\}$ where does $\Gamma$ fail to be l.h.c.? This is a question on page 60 in the book "Recursive Methods in Economic Dynamics" by Nancy L. Stokey and Robert E. Lucas, JR. with Edward C.Prescott. Can someone give a hint? Jackzhp (talk) 00:17, 10 August 2010 (UTC)[reply]

Yes, wikipedia is not a place you go to ask homework questions. If you insist, you might try the math help desk. 0¹⁸ (talk) 17:28, 29 August 2011 (UTC)[reply]

Closed domain

The current revision of the article claims: "For a correspondence Γ : A → B with closed values (i.e. Γ(a) - closed for all a in A), closed domain and compact range, to be upper hemicontinuous it is sufficient and necessary to have closed graph." However, it is not defined what closed domain means. (And this theorem is given here without a reference.) --Kompik (talk) 16:57, 20 June 2016 (UTC)[reply]

You're right. Added more details to this theorem and a reference for it on revision 730453369. Saung Tadashi (talk) 03:14, 19 July 2016 (UTC)[reply]

Upper and lower Vietoris topology

The link to the Vietoris topology only defines the Vietoris topology in general. It is not immediately clear how one would define an upper and a lower Vietoris topology based on the link. 120.51.141.38 (talk) 02:36, 19 May 2023 (UTC)[reply]