T1 space

Definition · applies to topological spaces · also called Fréchet space (in some texts), accessible space

Graduate (topology)

Consider a topological space $X$. The space $X$ is T₁ if every point of $X$ is a closed set.

Examples

Examples

• $\reals$ with the standard topology Undergraduate
• The line with two origins Graduate
  The line with two origins is T₁, because each point is closed.

Properties

• A space is T₁ if and only if, for every two distinct points, each has a neighborhood that does not contain the other.

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