In considering the Trinity, first it must be noted that God follows a supernatural logic, not an natural, earthly logic. For example, he is eternal, present everywhere in the universe. However, in regard to the persons of the Trinity being distinct, not separate, that is in regard only to function. Let me quote an outline I wrote--although it's special formating for reading comprehension and retention will not carry over to forum software. The entire article is at
http://www.loveofchrist.info/theology/trinity.html
from II. C.
"3. The distinction to be observed in the persons of the deity is in their function.
1. In regard to salvation, God, the Father, is the planner, and the Holy Spirit brings to completion the work that Christ, the uniquely related Son, initiates.
2. I Peter 1:2: Elect according to the foreknowledge of God the Father, through sanctification of the Spirit, unto obedience and sprinkling of the blood of Jesus Christ…  cf. God, the Father, as planner, Matthew 11:26; John 3:16; Romans 8:2; Ephesians 1:9: Christ as redeemer, Matthew 1:21; Romans 3:24,25; Galatians 4:4,5; I Corinthians 1:30; Ephesians 1:7; I Timothy 2:5: Holy Spirit as regenerator, John 3:5; 14:16,26; 16:7–11; Acts 1:8; Romans 5:5; 8:14,15; 14:17; I Corinthians 2:12,13; Galatians 5:22,23; Ephesians 1:13; Titus 3:5; I John 5:6.
3. In regard to creation, the first verses of Genesis again indicate that God, the Father, is the planner; Christ, as his word, initiates; and the Holy Spirit, hovering over an earth without form, brings the work to completion.
4. Scripture always represents the threefold division of God as persons, never as a force.
1. God as creator is Father. I Corinthians 8:6: But to us there is but one God, the Father, of whom are all things…, cf. Luke 3:38; Hebrews 12:9.
2. Jesus Christ is the Son, as heir of all things. Hebrews 1:2:…whom he hath appointed heir of all things, by whom also he made the worlds, cf. Ephesians 1:9–12; Colossians 1:13–18. As the son of man, Mark 6:3, and the Son of God, Matthew 17:5, Christ’s work on earth initiated a new humanity, Romans 5:18,19, who will be the citizens of his coming kingdom, Revelation chp.21.
3. The Holy Spirit proceeds from the Father and the Son as life giver. John 3:5,6:…Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of flesh is flesh; and that which is born of the Spirit is spirit. Romans 8:15–17:…ye have received the Spirit of adoption, whereby we cry, Abba, Father. The Spirit itself bears witness with our spirit, that we are the children of God: And if children, then heirs; heirs of God, and joint heirs with Christ…, cf. John 16:8–13; Romans 8:11; II Corinthians 3:17; Galatians 5:25; Ephesians 1:13,14; I Thessalonians 1:5.
"III. Scripture does not fully explain the complex existence of God, but it reveals his diversity to the extent that the Christian doctrine of his tri-unity may be defined formally. (Note item one of the Bibliography to reference the below cited page numbers)
1. The persons of God are distinct, not separate.
1. They do not exist alongside each other, “but in, through, and unto each other,†(p.296). “Neither space nor time nor anything else separates them,†(p.298).
2. The essence of deity is fully possessed by each person and by the tri-fold unity. Each person “is identical with the entire being, and equal to the other two persons taken together, or to all the three,†(p.302).
3. However, the persons of God “are not the three parts of one whole nor three names for one and the same object. The Father alone is Father, the Son alone is Son, the Spirit alone is Spirit,†(p.282).
4. We do not worship the Son and the Holy Spirit besides the Father, but in God we worship the Father, Son, and Holy Spirit.
5. The Holy Spirit is the name of the third person of the Trinity, although he is also the personal Spirit of the Father and the Son, Romans 8:9,27; just as all three persons have the same mind, will, energy, nature, and essence. The Word of God does not manifest the exact manner that three persons can have the same spirit.
6. However, Scripture never contradicts itself, stating in one passage that there is one God, and in other verses that there are three gods, but it consistently represents God as three in one."
In the above quote, in regard to the issue of this thread it is interesting that the three persons of God are represented as having one spirit. However, if the Trinity is rejected, then the concept of Christ changes, and the Christian religion is no longer valid. That's why opponents of Christianity so often begin their attack of the religion with the Trinity, because it depends on a supernatural logic from a supernatural existence, which people in a natural world can find difficult to comprehend. However, without the doctrine of the Trinity, the overall references to God in the Bible cannot be understood consistently--example, Zechariah 12:10:
And I will pour upon the House of David…the Spirit of grace and supplications: And they shall look upon me whom they have pierced, and they shall mourn for him….
Regards,
Paul