Surely it is important to note that the societies that have women going around topless are not Christian. Many are animist and tribal. Those who live in nudist colonies are not Christians either.
 I say that nudists are not Christians because modesty is commanded - nudity is not modest - therefore nudists are disobeying 1 timothy 2:9 which says; "women should adorn themselves in respectable apparel, with modesty and self-control." [/edit]
Why did God give Adam and Eve clothing?
God gave them clothing because of their sexual attraction.
 If clothing were for some reason other than sexual attraction, why would modesty be commanded? [/edit]
Mathew chapter 5 clearly states that a man who looks at a woman with lust has already committed adultery with her in his heart. Christians are commanded to purify their hearts (James 4:8) and avoid all forms of sexual immorality (Acts 15). Paul also commands modesty.
What is modesty? The fact that modesty is not explicitly defined means that every Christian must draw a line in the sand. Where that line is drawn exactly will vary slightly.
The case may be made that certain body parts are inherently sexual in nature. Proverbs 5:19 says "let her breasts (your wife's) satisfy thee at all times; and be thou ravished always with her love.." Since breasts are inherently sexual and women are to be modest, it would seem necessary that breasts should be covered.
One test for modesty is shame. Adam and Eve were ashamed when they realized that they were naked (Genesis 3). What caused Adam and Eve's shame?
"At that moment their eyes were opened, and they suddenly felt shame at their nakedness." Genesis 3:7 NLT
Sin + nakedness caused their shame. Sin tainted everything including sexual attraction. Adam and Eve's tainted sexual attraction caused shame. This is not to say that sexual attraction is inherently sinful - but rather that sin has often tainted it.
Nakedness and immodesty cause shame in our fallen world. People are ashamed to be seen in their underwear. Bikinis and underwear are virtually indistinguishable. Therefore bikinis are immodest. The culture (not the Bible) appears as the primary cause of Christians tolerating bikinis.