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Let $H^{\ast }$ denote the Čech cohomology or Alexander-Spanier cohomology.

Definition: (Tautness property of cohomology) Let $X$ be a paracompact Hausdorff space and $A$ be a closed subspace of $X$. If $H^{\ast }\left( A\right) =\underrightarrow{\lim }_{U\supset A}H^{\ast }\left( U\right) $, then $A$ is said to be tautly imbedded in $X$. Here $U$ runs over the neighborhoods of $A$ in $X$. If $A$ is tautly imbedded in $X$ for every closed subspace $A$, then the cohomology $H^*$ is said to be provide the tautness property.

Definition: (Continuity property of cohomology) Let $\left\{ X_{i}:i\in I\right\} $ be an inverse system of paracompact Hausdorff spaces and $X=\underleftarrow{\lim }X_{i}$. If $H^{\ast }\left( X\right) =% \underrightarrow{\lim }H^{\ast }\left( X_{i}\right) $, then $H^{\ast }$ is said to provide the continuity property.

In most places, it uses the continuity property instead of the tautness property. My question is that are these two concepts equivalent?

Edit: Capel [1] showed that tautness and continuity are equivalent for the Čech cohomology on compact pairs.

[1] C. E. Capel. Inverse limit spaces.Duke Math. J. 21(2): 233-245 (June 1954) https://projecteuclid.org/journals/duke-mathematical-journal/volume-21/issue-2/Inverse-limit-spaces/10.1215/S0012-7094-54-02124-9.short

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    $\begingroup$ @DaveBenson What I actually want to ask is that Alexander-Spanier cohomology has the tautness property, I wonder that it has the continuity property? It is true for an inverse system of compact Hausdorff spaces. It can be found in Spanier's book Algebraic Topology on page 319. I think it is correct for paracompact Hausdorff spaces, but I cannot find a complete source. $\endgroup$
    – Mehmet Onat
    Commented Apr 23 at 10:07
  • $\begingroup$ @DaveBenson I asked before at mathoverflow.net/questions/257588/…. I'm not exactly sure if it follows from Corollary 22 in Watanabe's article. $\endgroup$
    – Mehmet Onat
    Commented Apr 23 at 10:08
  • $\begingroup$ @DaveBenson Moreover, there exists a claim on page 384 in D. Anderson, W. Fulton. Equivariant Cohomology in Algebraic Geometry google.com.tr/books/edition/…. It is mentioned there that the bonding maps are onto. I couldn't find any information about this in Watanabe's article. $\endgroup$
    – Mehmet Onat
    Commented Apr 23 at 10:15

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