M.S. Comprehensive Examination Topics
Topology
2010



 ⁃    Topological Spaces
        ⁃    topology
        ⁃    open set
        ⁃    neighborhood
        ⁃    closed set
        ⁃    closure
        ⁃    interior
        ⁃    boundary
        ⁃    limit point
        ⁃    Hausdorff topology
        ⁃    separable space
        ⁃    base
        ⁃    countable base
        ⁃    subspace topology
        ⁃    order topology
        ⁃    metric topology
        ⁃    finite product topology 
 ⁃    Continuous Functions
        ⁃    continuous mapping
        ⁃    open map
        ⁃    closed map
        ⁃    quotient topology
        ⁃    identification map   
    ⁃    Connectedness
        ⁃    connected space
        ⁃    continuous image of a connected space
        ⁃    path connected space
        ⁃    continuous image of a path connected space
        ⁃    product of two path connected spaces
        ⁃    product of two connected spaces
        ⁃    locally connected space
        ⁃    locally path connected space
        ⁃    component
        ⁃    path component   
    ⁃    Compactness
        ⁃    compact spaces
        ⁃    tube lemma
        ⁃    finite intersection property
        ⁃    closed and bounded subsets of Euclidean spaces
        ⁃    cantor sets
        ⁃    limit point compactness
        ⁃    Lebesgue number of a covering
        ⁃    local compactness
        ⁃    one-point compactification
        ⁃    continuous image of a compact space
        ⁃    the product of two compact spaces
        ⁃    uniform continuity and compactness
 -    The fundamental group
         - Covering spaces
         - The fundamental group of the circle
         - Retractions and deformation retracts
         - Brouwer fixed point theorem for the disc, Fundamental Theorem of Algebra and other applications

    

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