Q:

Java core API part of JDK or JRE?

I want to know whether the official Java core API is part of JDK or JRE. 

According to my understanding of Java, I think that the core API should be part of the JRE, as the application should be runnable without the additions provided by the JDK. 

I also found some information on the web that says the 3 basic components of the JDK are,  

Java Compiler
JVM
Java API

(As the JRE is the implementation of JVM, does the Java API belong to JRE or JDK?)

Edit: actual question: Is the Java API included in JRE?

A:

Your facts are a bit inaccurate.

I also found some information on the web that says the 3 basic components of the JDK are: 1. Java Compiler, 2. JVM, 3. Java API. 

That's not a good summary.  There are certainly other things in a JDK apart from that.

As the JRE is the implementation of JVM

In fact the JRE includes an implementation of the JVM.   And it also includes the core Java APIs, and a few other things.
As Holger points out, a JDK consists of a JRE plus some additional Java development tools.  Or as he succinctly puts it:

"JDK = JRE + development tools"

So to answer your question:

does the Java API belong to JRE or JDK?

The Java APIs are included in both a JRE distribution or JDK distribution, but the phrase "belong to" doesn't have much meaning in this context.

(Now one could debate whether a JDK "contains" a JRE (or not), and whether the JRE "contains" the Java APIs.  But to be honest it is a pointless debate.  What really matters is that the Java APIs are present in both kinds of distribution.  
Also, this is substantially moot in Java 11, since Oracle no longer provides JRE distributions for either Oracle or OpenJDK Java.  It is now JDK only.)  

