Leave $ X, Y $ Be two Hausdorff spaces and $ F: X to 2 ^ Y $ Be a multi-valued mapping We say that $ F $ is semicontinuous lower to $ x_0 in X $ yes for each $ y_0 in F (x_0) $ and any neighborhood $ U in mathcal N (y_0) $, there is a neighborhood $ V $ of $ x_0 $ such that
$$
F (x) cap U neq emptyset text {for all} x in V.
$$
In Aubin and Cellina. Differential inclusion, it is stated that the property is equivalent to
given any generalized sequence $ x_ mu a x_0 $ and any $ y_0 in F (x_0) $, there is a generalized sequence $ y_ mu in F (x_ mu) $ such that $ y_ mu a y_0 $.
I guess they meant net When they said generalized sequence.
I do not think this statement is trivial at all. In fact, I found proof of this from a different source that I think is not valid.
Leave $ (x_ mu) _ { mu in mathcal I} $ be a convergent network to $ x_0 $, the aforementioned test builds a network. $ y: D to Y $ of a directed set $ D subset mathcal I times mathcal N (y_0) $ defined by
$$
D: = {( mu, V): F (x_ mu) cap V neq emptyset }
$$
by choice
$$
y _ { mu, V} in F (x_ mu) cap V.
$$
It is not difficult to verify that $ (y _ { mu, V}) _ {(, V) a D} $ converges to $ y_0 $. However, your last step to find a map. $ sigma: mathcal I to D $ compose with $ and $ It seems invalid, since they leave $ sigma $ be an arbitrary section of $ pi: D to mathcal I $, where $ pi $ is the projection $ pi ( mu, V) = mu $.
However, I looked around and found an old paper from Frolík, On the topological convergence of sets., since 1960. It contains a statement that implies a weaker result (but it seems more reasonable) that could be summarized as follows.
Leave $ (M_ mu) _ { mu in mathcal I} $ be a network in $ 2 ^ Y $. So $ y_0 $ is in $ liminf M_ mu $ Yes and only if there is a network. $ pi: mathcal I & # 39; to mathcal I $ which residual in $ mathcal I $ and points $ y_ lambda in M _ { pi ( lambda)} $ such that $ (y_ lambda) _ { lambda in mathcal I & # 39;} to y_0 $.
The test is almost identical to the previous one, but stopped by letting $ mathcal I & # 39; = D $, defined as in the previous test, and concluded the theorem without having to invoke a section $ sigma $ of $ pi $.
Question: Is the affirmation of Aubin and Cellina true? If so, where can I find a valid proof of that?
