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Steal my completeness and I'm still productive.

Steal my productivity and I'm still normal.

Steal my normality and I'm still metric.

Steal my metric and I'm still open.

What am I?

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    $\begingroup$ (rot 13) Guvf zhfg or fbzr xvaq bs zngurzngvpny fcnpr evtug? What happens if I take away your openness? :-P $\endgroup$
    – Eric
    May 5, 2023 at 17:11
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    $\begingroup$ @Eric Steal my openness and I'll be killed...? $\endgroup$ May 5, 2023 at 21:21

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I think you must be talking about a

Hilbert space

Steal my completeness and I'm still productive.

Every Hilbert space is complete meaning that every Cauchy sequence has a limit in the space. Even if we remove the restriction of completeness to define a much larger set of topological spaces we still have a space which has an inner product.

Steal my productivity and I'm still normal.

If we then remove the inner product from our topological space we are still left with a space on which a norm is defined - a normed vector space.

Steal my normality and I'm still metric.

If we were to remove the norm from our space we would be left with a metric space which is just a topological space endowed with a notion of distance.

Steal my metric and I'm still open.

If we remove the metric from our space we are left with a pure topological space. Topological spaces are often defined via a collection of open sets with certain properties and, within this construction, the entire space itself is also an open set.

This diagram I took from Wikipedia explains it quite well

enter image description here
with the caveat that Hilbert spaces constitute another circle contained within inner product spaces.

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