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nLab > nLab General Discussions: injective objects in topological spaces

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2016
    • (edited Jun 7th 2016)

    Somebody has contacted me by email, regarding an issue he sees with the entry injective object in the section In topological spaces (which was added by Todd in revision 37). He writes:

    The injectives in Top are indeed the indiscrete spaces.

    But the injectives in Top0 are the terminal spaces (not the continuous lattices with the Scott topology).

    The above are the injectives wrt to monomorphisms.

    Now, if you speak of injectives with respect subspace embeddings (=homeomorphic embeddings), then, in Top0, the injectives are precisely the continuous lattices endowed with the Scott topology.

    And, in Top, the injective wrt to subspace embeddings are the continuous “pre-lattices” under the Scott topology, we “pre” refers to the generalization of lattice in which the order is merely a pre-order, rather than a partial order.

    • CommentRowNumber2.
    • CommentAuthormartinescardo
    • CommentTimeJun 7th 2016
    I've fixed this now.
    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 26th 2016

    Thanks to Martín for catching this. It seems clear that what threw me off was the fact that such spaces, coming from Scott topologies on continuous lattices, are typically called “injective spaces”.

    I added a terminological note to that effect.

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