How did America after the Spanish-American war act as a "great power" in world affairs, especially in Asia?

After the Spanish-American War, the United States asserted
its power more in various places around the world.  If you want to talk mainly about
Asia, let us look at four things:

• The taking of
  the Philippines.  The US took the Philippines, first from the Spanish and then by
  fighting Filipino independence fighters.  It then used the Philippines as a colony and a
  military base.


• The "Open Door" in China.  In this case,
  the US demanded equal access (with European countries and Japan) to
  China.


• The Great White Fleet.  The US sent a fleet around
  the world, including to Asia, as a way of showing its
  power.


• Theodore Roosevelt's mediation of the
  Russo-Japanese War.  TR acted as a mediator to put an end to this war between Russia and
  Japan.

In all of these ways, the US was showing
that it felt that it was strong enough to take a leading role in Asia.  It was acting as
a country that deserved to be involved in the affairs of this area, even though it was
far from the continental US.