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If by a normal character you mean a normal morphism of C*-algebras A→C, then every commutative von Neumann algebra canonically decomposes as a product of its atomic and diffuse parts, the atomic part …

Corollary 4.1.(i) in Johnstone's book Stone Spaces states that the category of realcompact spaces is dual to the full subcategory of the category of commutative rings consisting of rings of the form C …

To supplement Mark Meckes' answer let me point out that all of your five examples are L^p-spaces of the corresponding von Neumann algebras: bounded functions on the disjoint union of n points for R^ …

Any presentation of a given space as a CW-complex immediately gives rise to a presentation of the fundamental groupoid, and hence also the fundamental group. Specifically, given such a presentation a …

Perhaps it makes sense to mention this example: The category of measurable spaces is equivalent to the category of hyperstonean topological spaces and hyperstonean maps between them. To construct a m …

There are at least three different answers that can be given to this question, and in all three interpretations the answer essentially states that all maps are “open”, for the appropriate analogue of …

The spectrum of $L^\infty(R)$ is the hyperstonean space associated with the measurable space R. More information can be found in Takesaki's Theory of Operator Algebras I, Chapter III, Section 1.

Without loss of generality we can assume that the support of the measure equals $X$ (i.e., the measure is faithful), because we can always pass to the subspace defined by the support of the measure. T …

To show that $f$ is a covering map, pick any point $n∈N$. Choose an open neighborhood $U$ of $n$ that is diffeomorphic to ${\bf R}^n$. It suffices to show that the restriction $g\colon V\to U$ of the …

The easiest way to construct an explicit contracting homotopy is to observe that EG is the geometric realization of the nerve of the groupoid G//G, which has G as its set of objects and exactly one mo …

In this answer, Topos is interpreted as a 2-category. (As a side remark, the 1-category of toposes does not make sense until one picks a specific model for toposes and geometric morphisms, and differe …

Any compact Hausdorff space $X$ is a Baire space: if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets, also known as a set of first category), then $X$ is empty. I am i …

Chris Douglas has a nice short discussion of homotopy limits in his text “Sheaves in homotopy theory” (Chapter 5 of Topological modular forms).

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a c …

So my first question is, can a “diagram of spaces over a CW complex” be a manifold? If so, under what conditions? Any smooth manifold is homotopy equivalent to the homotopy colimit of a diagram of c …