  The field content of the two dimensional string theory consists of the
dynamical tachyon field and some nonpropagating fields which consist in the
topological sector of this theory. We propose in this paper to study this
topological sector as a spacetime gauge theory with a simple centrally extended
$w_\infty$ algebra. This $w_\infty$ algebra appears in both the world sheet
BRST analysis and the matrix model approach. Since the two dimensional
centrally extended Poincar\'e algebra is naturally embedded in the centrally
extended $w_\infty$ algebra, the low energy action for the metric and dilaton
appears naturally when the model is truncated at this level. We give a
plausible explanation of emergence of discrete states in this formulation. This
theory is again the effective theory at zero slope limit. To include higher
order $\alpha'$ corrections, we speculate that the whole theory is a gauge
theory of a deformed $w_\infty$ algebra, and the deformation parameter is just
$\alpha'$.
