Glycolysis only in organisms that breathe in oxygen

False.

Glycolysis is the 1st metabolic step for both aerobic and anaerobic metabolism. The process of glycolysis takes 1 glucose molecule and 2 ATP molecules, then produces 2 pyruvate molecules and 4 ATP molecules for a net gain of 2 ATP molecules. The ATP is used by the organism for immediate energy and the pyruvate is further metabolized using either an aerobic or anaerobic metabolism. If anaerobic, the pryruvate is fermented into either ethanol and carbon dioxide, or into lactic acid. The aerobic pathway is a long chain of reactions that eventually produces an additional 28 or 30 ATP molecules for the organism's energy needs.

So as you can see, both aerobic (oxygen using) and anaerobic organisms perform glycolysis.

Glycolysis is a process by which glucose is broken down into pyruvic acid in the cytoplasm of the cell. this process will occur in presence or absence of oxygen and therefore will not only occur in organisms that breath oxygen. in aerobic organisms, after the pyruvic acid has been formed it enters into mitochondrion where it undergoes a series of reaction to release more molecules of ATP. the final step in mitochondrion involves a reaction between oxygen and hydrogen ions to produce water as one of final products of respiration. in anaerobic organisms the pyruvic acid produced by the glycolysis is converted to molecules of lactic acid as the final product of respiration.