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Version 3 |
| A topological space $X$ is {\em compact} if, for every collection $\{U_i\}_{i \in I}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection $\{U_{i_j}\}_{j=1}^n$ whose union is also $X$. |
A topological space $X$ is {\em compact} if, for every collection $\{U_i\}_{i \in I}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection $\{U_{i_j}\}_{j=1}^n$ whose union is also $X$. |
| A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space. |
A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space. |
| {\bf Note:} Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. |
{\bf Note:} Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. |
