Well it was pretty much a world war since about every other country that mattered at the time was at war, except for the US/Canada. Note that the US was anyway part of the War efforts as they were supporting Great Britain, too.
The war in the west didn't start until May 1940, when Germany invaded France. Shortly after that (June 1940) the air battle over Britain started, and at least there was a Canadian participation: https://en.wikipedia.org/wiki/Non-British_personnel_in_the_R...
Edit: Canada already sent infantery units to Britain during the Phoney War (that was the period until the war in the west started:
"The Royal Air Force dropped propaganda leaflets on Germany and the first Canadian troops stepped ashore in Britain, while western Europe was under a period of uneasy calm for seven months.[8]"
https://en.wikipedia.org/wiki/Phoney_War
> The war in the west didn't start until May 1940, when Germany invaded France.
To be picky, the Battle of the Atlantic started the same day as the British and French declaration of war, with the sinking of the SS Athenia. And the Norwegian Campaign began the month before the invasion of France.
But the wars weren't connected: aside from some skirmishes between the USSR and Japan (see https://en.m.wikipedia.org/wiki/Soviet%E2%80%93Japanese_bord...), there was a British/French/Russian/German war going on in Europe, and a Sino-Japanese war in Asia. It wasn't until Pearl Harbor, and the simultaneous Japanese declaration of war on the European Allied powers, that the two wars became one.
The war was not just going on in Europe. North Africa was also heavily involved in skirmishes between Allies and the Axis.
As for the connection there was only one country that was fighting on two fronts at the same time: the US. Limiting the WW2 definition to when the US started to enter the War is reductive, I think.
"In effect, Australia fought two wars between 1939 and 1945[2] – one against Germany and Italy as part of the British Commonwealth's war effort and the other against Japan in alliance with the United States and Britain. While most Australian forces were withdrawn from the Mediterranean following the outbreak of war in the Pacific, they continued to take part in large numbers in the air offensive against Germany."
