There are only certain image formats that support transparency.
If you save your image as a PSP image it will retain all layers and transparency as is. Of course this prevents you from actually using it in anything other than PSP.
You can however use the export feature, to preserve the transparency of your image. PSP will allow you to export into either Gif, Jpeg or PNG. Jpeg does not support transparencies so you are limited to either Gif or PNG. Gif is limited to only 256 colors. So your best bet to preserve transparency, and still look nice, is PNG. PNG will display in almost any place you wanted to display and is web supported by many browsers.
to use the export feature, go to them File menu, and depending on your version of PS will either say Import/Export, or just Export. then choose the PNG optimizer option.
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Ignorance is not necessarily Bliss, case in point:
Unknown has caused an Unknown Error on Unknown and must be shutdown to prevent damage to Unknown.