## Non-Hausdorff Ascoli theory |

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### Contents

Introduction | 5 |

Topological theory for multifunctions | 14 |

Uniform theory for multifunctions | 27 |

1 other sections not shown

### Common terms and phrases

Ascoli theorems bourhood Ck(X closed in 0(X compact neighbourhood compact subset compactness of F conditions are sufficient continuous multifunctions continuous on compacta Corollary denote the rp-closure equicontinuous on compacta exists a neighbourhood F in YmX)0 F is closed F is compact F is equicontinuous F is jointly F is strongly F is Tc-compact F satisfies G family of multifunctions family of point-compact fc-continuous fc-space finite intersection property finite sequence Gale's theorem Hausdorff or regular implies jointly continuous k-space Kelley-Morse Lemma Let F denote Let xeX locally compact space logical space lower semi-continuous neighbourhood topology non-empty sets nuous open neighbourhood open subset point-compact multifunctions pointwise compact Proof quasi-uniform space rc on F rc-closed regular space rp-closure of F rp-compact Smithson 32 subbasis subfamily subset of F suffice to show topo topological space Tychonoff theorem U[fx uniform convergence uniform space