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 FrolΓk, *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

residualin $ mathcal I $ and points $ y_ lambda in M ββ_ { 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?