Yes, I know some will quote a verse or two they say supports the idea but those quoted verses have nothing to do with any imagined “succession”.
We know that the Christians in that first century gathered together in worship and teaching as a community. They shared their resources and ministered to the poor. The actual Apostles went throughout the known world preaching the gospel of Jesus Christ and ordaining elders in each community to carry on the teachings they handed down to them. Rather than there being some kind of "Apostolic Succession" where an Apostle could name his own successor and give to him the mantle of Apostle to then pass down to whomever he chose, the succession was one of DOCTRINE and it was in this that the authority could be claimed. And, just because a church was established by an Apostle, did not guarantee that that church would ALWAYS continue to be faithful to the Apostolic teachings. In fact, we learn from http://www.newadvent.org/cathen/01634a.htm, that:
All the individual orthodox churches could, in a sense, be called Apostolic Churches, because they were in some more or less mediate connection with the Apostles. Indeed, that is the meaning in which Tertullian sometimes uses the expression Apostolic Churches (De Praescriptionibus c. xx; Adversus Marceonem, IV, v). Usually, however, especially among the Western writers, from the second to the fourth century, the term is meant to signify the ancient particular Churches which were founded, or at least governed, by an Apostle, and which, on that account, enjoyed a special dignity and acquired a great apologetic importance. To designate these Churches, Irenaeus has often recourse to a paraphrase (Adv. Haer., III, iv, 1), or he calls them the "oldest Churches". In the writings of Tertullian we find the expressions "mother-Churches" (ecclesiae matrices, originales), frequently "Apostolic Churches" (De Praescriptionibus, c. xxi).
At the time of the Christological controversies in the fourth and fifth centuries some of these Apostolic Churches rejected the orthodox faith. Thus it happened that the title "Apostolic Churches" was no longer used in apologetic treatises, to denote the particular Churches founded by the Apostles. For instance, Vincent of Lérins, in the first half of the fifth century, makes no special mention in his "Commonitorium" of Apostolic Churches. But, towards the same epoch, the expression "the Apostolic Church" came into use in the singular, as an appellation for the whole Church, and that frequently in connection with the older diction "Catholic Church"; while the most famous of the particular Apostolic Churches, the Roman Church, took as a convenient designation the title "Apostolic See" (Vincent of Lérins's Commonitorium, c. ix). This last title was also given, though not quite so often, to the Antiochian and to the Alexandrian Church.
Many of the Roman Catholics on this forum absolutely insist that "their" church is the same church that Christ established in the first century. They claim this Apostolic character is EXCLUSIVELY theirs, but they must be held to the same measuring rod as any Christian church should be and that is by Scripture. God ensured that we would have an objective and static resource in order to know the SAME doctrines the Apostles were given and so that any claims of apostolicity could be determined by it - the New Testament Scriptures. If what a church teaches doesn't square with the Bible, then it doesn't matter what the sign says outside the doors. Christianity has a rule of faith. It doesn't change and it is found in Scripture.