We present a particular case of the formation of a penumbra sector around a developing sunspot in the active region NOAA 12574 on 2016 August 11 by using the high-resolution data observed by the New Solar Telescope at the Big Bear Solar Observatory and the data acquired by the Helioseismic and Magnetic Imager and the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory satellite. Before the new penumbra sector formed, the developing sunspot already had two umbrae with some penumbral filaments. The penumbra sector gradually formed at the junction of two umbrae. We found that the formation of the penumbra sector can be divided into two stages. First, during the initial stage of penumbral formation, the region where the penumbra sector formed always appeared blueshifted in a Dopplergram. The area, mean transverse magnetic field strength, and total magnetic flux of the umbra and penumbra sector all increased with time. The initial penumbral formation was associated with magnetic emergence. Second, when the penumbra sector appeared, the magnetic flux and area of the penumbra sector increased after the umbra's magnetic flux and area decreased. These results indicate that the umbra provided magnetic flux for penumbral development after the penumbra sector appeared. We also found that the newly formed penumbra sector was associated with sunspot rotation. Based on these findings, we suggest that the penumbra sector was the result of the emerging flux that was trapped in the photosphere at the initial stage of penumbral formation, and when the rudimentary penumbra formed, the penumbra sector developed at the cost of the umbra.