Here are examples of the pith topi or 'Bombay Bowler.' One belongs to a Seaforth Highlander and the other is a British made example purchased by a US Serviceman with the US Officer's Insignia attached. See if you can guess which is which?
The US Serviceman's topi has a blue band at the top of the puggaree or ' pagri' which probably indicated that it was initially issued to the Royal Navy. Here is a second Sun Helmet, pith topi (Bombay Bowler) which was made in Madras in India and was worn by a Seaforth Highlander. This one has a regimental batch on the left side mounted against the Seaforth Highlander tartan with a 'hackle' of red and white feathers.
"An original private purchase solar topee 'pith helmet' which belonged to an officer of the Seaforth Highlanders during the Second World War. This helmet most likely belonged to an officer of the 1st battalion of the Seaforth Highlanders and was most likely purchased around 1942 when the regiment joined the 23rd Infantry Division in India before moving on to battle in Burma. It was purchased by the officer in Madras and is a typical style worn by officers during this period. Solar topees were still a common sight in the early part of the war but not seen much after the fall of Singapore. The helmet is made from shola pith which is covered in a khaki colored cloth. A pugree also spelt as ‘puggaree’ and ‘pagri’, is fitted around the circumference of the helmet and a leather chin strap is also present. To the side of the helmet a Seaforth Highlanders cap badge has been fitted on a tartan backing and a red and white feature is fitted above this. The interior of the helmet features a liner which has a leather sweatband. The original label is fitted to the inside of the crown and bears the maker's name of 'M. Mohamed Eyan Sahib'. The address of 28 Moore Market, Madras is also present as are the words 'Real Pith Sola Hat'."
And just for fun, here is the early version from the Movie Zulu
, which depicted the 1879 Battle of Rorke's Drift. While the regiment is heavily portrayed as Welsh in the movie, it was historically the 24th (2nd Warwickshire) Regiment of Foot, which was an English county regiment with its recruiting base in the Midlands, though it later became the South Wales Borderers in 1881. At Rorke's Drift, the garrison was actually a diverse mix of English (49), Welsh (32), and Scottish, Irish, and overseas-born soldiers. [1, 2, 3] 
Michael Caine as Lt. Gonville Bromhead. Zulu Film (1964).