くろくまさん、neverbirth2848-san, マイケルさん、
It's probably how you consider "topic" in that sentence's case. And also we might need to
classify normal "topic" as "sentence topic".
The main topic (and the one continues to be the sentence topic until the end of that sentence) is 私（は）
although it's not clearly mentioned. And regarding the rest, I pretty much agree with Michael-san's explanation.
I'll try to give my insights/analysis on this issue.
The particle は is anyway what's called "topic marker", but in my sentence, there are (supposed to be) three
はs in it:
私は、りんごは好きですが、みかんは嫌いです。
りんご and みかん can be also considered as "topic"s and probably better call them "sub-topics".
The reason for making any word "topic" is because we need to clarify what we're talking about (basically), right?
If this was only about simple likes and dislikes, it should be
私は、りんごが好きで、みかんが嫌いです。
In this sentence, clearly the topic is 私 and, in other words, 私 is the topic you have to clarify and/or emphasise.
Just imagine it's the Japanese class and students have to tell likes and dislikes in Japanese.
Everyone can use this simple sentence, just replacing the fruits' names.
In such a situation, you shouldn't drop 私は.
One of the biggest differences between は and が is "until where the marked word/noun has influence".
Subject part needs the corresponding predicate.
This relation of "subject-predicate" can be found in a very close place in a sentence if が is used to mark
the subject. And soon after the marked subject meets its predicate, the function as "subject" is over.
On the other hand, if は is used and marked word is "topic", the predicate(s) can be found anywhere in
that sentence and this topic continues until the end of that sentence.
Now going back to 私は、りんごは好きですが、みかんは嫌いです。 (A) and
私は、りんごが好きで、みかんが嫌いです。(B),
the sentence topic (i.e. real topic OR main topic) of the sentence A is 私 and entire sentence is about 私.
In order to tell what "I" like and dislike in a clearer way with certain emphasis, this 私 decided to replace が
with は and made those two fruits "topic" as well. When two similar parts shares the same concept or idea,
it makes the contrast. Making contrast, you're somewhat spot lighting the "words" (りんご and みかん in this case).
At the same time, the part りんごは好きです（が） part remains influencing the rest of the sentence
(by making contrast with みかん) and the same goes to みかんは嫌いです part.
In the sentence B, on the other hand, as soon as 私 says りんごが好き, this discussion of "liking apples" is over
and 私 move on to the next discussion -"disliking oranges". Those two parts are practically "independent" and
doesn't affect each other in any way.
So, to answer to Kurokuma-san's question (whether or not there can be two or more topics in a sentence),
I'd probably say "yes", but it's just not the same level as sentence topic.
Hope this helps
Natsuko (奈津子),
Team JapanesePod101.com
