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This paper studies the space $X_g\subset \mathcal{M}_g$ consisting ofsurfaces with filling systoles and its subset, critical points of the systolefunction. In the first part, we obtain a surface with Teichm\"uller distance$\frac{1}{5}\log\log g$ to $X_g$ and in the second and third part, prove thatmost points in $\mathcal{M}_g$ have Teichm\"uller distance $\frac{1}{5}\log\logg$ to $X_g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}-\sqrt{7\log\logg})$ respectively.Therefore we prove that the radius-$r$ neighborhood of $X_g$is not able to cover the thick part of $\mathcal M_g$ for any fixed $r>0$. Inlast two parts, we get critical points with small and large (comparable todiameter of thick part of $\mathcal M_g$) distance respectively.

Shape of filling-systole subspace in surface moduli space and critical points of systole function