Abstract: In this note, the class of cut^*spaces are introduced.
A connected topological space X is called a cut^* space, if $X\setminus \{x\}$ is connected for any $x\in X$, and $X\setminus \{x,y\}$ is not connected for any distinct points $x,y\in X$. The properties of cut^* spaces are discussed. The authors show that if X is a cut^* spaces, then $\{x\}$ is open or closed for any $x\in X$, and $X$ has infinite closed points. The authors also discuss the properties of some special cut^* spaces, which satisfy that $X\setminus \{x\}$ is COTS for any $x\in X$. The authors show that there doesn't exisxt a cut* space $X$, satisfying that $X\setminus \{x\}$ is a irreducible cut space for any $x\in X$. In the last part of the note, the authors show that $X$ is compact or Lindel\"of, if $X$ is a cut^* space and $X\setminus \{x\}$ is LOTS for any $x\in X$.

Key words: cut space, cut^* space, LOTS, Khalimsky line