A solar-heating system having an inflatable solar collector connected to a housing that can be installed, e.g., in a sash window of a house. The housing has an air blower that draws air from the interior of the house and directs it into the inflatable solar collector, which is positioned outside the window. The collector has a flexible permeable membrane that serves as a transpired absorber of solar radiation. The air flowing through the collector is heated when it percolates through the membrane exposed to solar light. The heated air is returned back to the interior of the house, thereby providing space heating. In certain embodiments, in addition to serving as a transpired absorber, the membrane can also serve as an air filter that removes particles, odor-causing pollutants, and/or allergens from the drawn air.