TRACON Controller described an airspace incursion event listing the facility policy of waiting until traffic builds and controllers request for help before securing additional staffing.

Narrative

I was working Sectors X and Y combined (on Y) when I was approached by a developmental Controller that informed me that they would be starting to train on these positions. When the OJT-I arrived at the position; I proceeded to give a briefing. As I was conducting my 2 minute post brief observation of the positions; it was decided to split the positions for training because the IFR traffic was starting to build. I then proceeded to open up X while the OJT-I and developmental worked X and Y. After working out a situation; the trainee then took over X while I took over Y. Within 1 minute of assuming responsibility for Y; an aircraft called on my frequency wanting to know if they should switch to Tower frequency. It was then; that I observed a primary/intermittent beacon target approximately 4 miles east of the airport. The developmental had issued a squawk code to this aircraft; but the aircraft was squawking intermittently. This information was not relayed to me before I took over the position (Y). This aircraft had entered inner Charlie airspace without a hand off being completed from Approach to Tower. Recommendation; our approach procedures are designed based on a 2 sector operation. Thus; Sectors X and Y need to be opened everyday between the hours of 9 am and 5 pm when all of the VFR traffic is prevalent. If these 2 positions had already been de-combined; this situation would not have occurred. This facility has the mindset that it is better to wait until the traffic builds; then wait for someone to request the sectors be split; then try to find someone to open up the position. By this time it is usually too late.