Every leaf in a lulav is comprised of two leaflets joined together by the posterior edge. The place where they are joined is called the “tiyomet,” because it makes each pair of leaflets resemble conjoined twins (“te’omim”). As the lulavim continue to grow and develop into ḥariyot, the tips of the leaves open up. The first to open is the central, uppermost leaf in the lulav. Since the leaves of the lulav are naturally closed because they are connected at the tiyomet, when the tiyomet opens, the lulav is considered deficient, not whole. Thus, even though it still looks like a lulav, it is disqualified for use on the first Yom Tov, as all the species must be whole. However, it may be used on subsequent days.
The poskim disagree about how to define a split tiyomet. Some say the lulav is invalid only if most of the tiyomot of most of the leaves are split (Rif and Rambam). Others say the law of the split tiyomet applies only to the upper, central leaf, since it is the most prominent and discernible of the leaves; if most of this tiyomet is split, the lulav is invalid (Ge’onim and Ran). The halakha in practice follows the latter view: on the first Yom Tov, if most of the central tiyomet is split, the lulav may not be used. In truth, in the vast majority of lulavim, most of the central leaf is closed, so almost all lulavim are kosher for use even on the first Yom Tov. Some are meticulous to avoid, le-khatḥila, a lulav whose central tiyomet has even a small split. However, there are very few lulavim where the tiyomet is completely closed, and the more developed and beautiful the lulav, the more likely that a bit of its uppermost leaf will be open. It would seem preferable to use a large, beautiful lulav with a slightly open central leaf than a small, shriveled lulav with a closed upper leaf. If one is concerned that the central leaf will continue to open and be mostly open by the first Yom Tov, he may glue it together to prevent its disqualification.
If there are two leaves at the central tip of the lulav, the tiyomet of each one has to be mostly closed. However, if these two leaves separate from one another, the lulav is kosher since they are two different leaves (MB 645:15; Pri Megadim, Mishbetzot Zahav ad loc. 4).
Some prefer to use a lulav which has “korei,” a flaky brown membrane that keeps the leaves together. In their view, as long as there is a korei on the leaves, they are considered closed. However, others say that le-khatḥila it is preferable to take a lulav without a korei, for two reasons: First, they maintain that the korei does not ensure that the central leaf is closed. True, it is not necessary to worry that it is mostly split underneath the korei, because that is extraordinarily rare, but it is possible that a small split is hidden underneath, which is not mehudar according to some. Second, Ashkenazic custom is to shake the lulav in such a way that it rustles. When a lulav has korei, this cannot be done.
In the opinion of Bikurei Yaakov 645:9, when the middle leaf is closed by the korei, it is considered closed even if it turns out after the korei is removed that it was open. From this perspective, a lulav with korei is mehudar. This is the ruling of R. Mordechai Eliyahu. However, Ma’amar Mordekhai 645:4 maintains that having korei does not mean the lulav is considered closed. Many defer to this opinion. Therefore, those who want to ensure that the lulav is closed according to all opinions prefer one without korei (Ḥazon Ish; Ḥazon Ovadia; Piskei Teshuvot 645 n. 13). For those who follow Ashkenazic custom, there is another reason that a lulav without korei is better: it rustles when shaken (Bikurei Yaakov 645:2). However, according to SA (651:9), there is no need for it to rustle.