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ebola1717
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"Cauchy" Sequences in General Topological Spaces
Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every neighborhood around the limit, the sequence is eventually entirely contained in that neighborhood. But this requires the sequence converges to a point in the space. For incomplete spaces, (does this idea exist in general topological spaces?) not all sequences that get close converge in the space.
For Cauchy sequence, then, it sounds like we want that for sufficiently large N and n>N, if m>n, then a_m is contained in a neighborhood around a_n, and that these neighborhoods get smaller, in some sense. My thought process lead me to believe we might be able to define "smaller" by strict inclusion, but it seems like the neighborhoods can stay pretty large in this case. In any case, is there a way we can define an analogous notion to Cauchy sequences in general topological spaces?
The motivation for this was that I was thinking about the definitions of compactness, specifically whether sequential compactness was equivalent to limit point compactness and normal compactness in topological spaces with weaker separation axioms than metrizability. I've been lead to believe the answer is no, and this seemed like a logical place to start.
Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every neighborhood around the limit, the sequence is eventually entirely contained in that neighborhood. But this requires the sequence converges to a point in the space. For incomplete spaces, (does this idea exist in general topological spaces?) not all sequences that get close converge in the space.
For Cauchy sequence, then, it sounds like we want that for sufficiently large N and n>N, if m>n, then a_m is contained in a neighborhood around a_n, and that these neighborhoods get smaller, in some sense. My thought process lead me to believe we might be able to define "smaller" by strict inclusion, but it seems like the neighborhoods can stay pretty large in this case. In any case, is there a way we can define an analogous notion to Cauchy sequences in general topological spaces?
The motivation for this was that I was thinking about the definitions of compactness, specifically whether sequential compactness was equivalent to limit point compactness and normal compactness in topological spaces with weaker separation axioms than metrizability. I've been lead to believe the answer is no, and this seemed like a logical place to start.