  Crystalline symmetries have played a central role in the identification of
topological materials. The use of symmetry indicators and band representations
have enabled a classification scheme for crystalline topological materials,
leading to large scale topological materials discovery. In this work we address
whether amorphous topological materials, which lie beyond this classification
due to the lack of long-range structural order, exist in the solid state. We
study amorphous Bi$_2$Se$_3$ thin films, which show a metallic behavior and an
increased bulk resistance. The observed low field magnetoresistance due to weak
antilocalization demonstrates a significant number of two dimensional surface
conduction channels. Our angle-resolved photoemission spectroscopy data is
consistent with a dispersive two-dimensional surface state that crosses the
bulk gap. Spin resolved photoemission spectroscopy shows this state has an
anti-symmetric spin texture resembling that of the surface state of crystalline
Bi$_2$Se$_3$. These experimental results are consistent with theoretical
photoemission spectra obtained with an amorphous tight-binding model that
utilizes a realistic amorphous structure. This discovery of amorphous materials
with topological properties uncovers an overlooked subset of topological matter
outside the current classification scheme, enabling a new route to discover
materials that can enhance the development of scalable topological devices.
