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A simple example of a ring that is an $A$module but not an $A$algebra?
I'm not sure why you say $\mathbb{R}$ is a $\mathbb{C}$vector space but not a $\mathbb{C}$algebra; any embedding of $\mathbb{C}$ in $\mathbb{R}$ makes $\mathbb{R}$ a $\mathbb{C}$algebra. In your situation, $B$ will be an $A$algebra iff the action of $A$ on $B$ commutes with the action of $B$ on itself. Equivalently, $B$ is an $A$algebra iff the action of $a\in A$ coincides with multiplication by $a\cdot 1_B$ in the ring structure of $B$. 
Sep 9 
answered  Completion of the set of subsets with half volume. 
Sep 4 
awarded  Enlightened 
Sep 4 
awarded  Nice Answer 
Sep 3 
answered  Haar Measure on Locally Compact Semigroups 
Sep 2 
revised 
Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
edited tags 
Sep 1 
revised 
Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{1}$ continuous at $(1,1)$
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Aug 27 
comment 
Clutching functions and Classifying maps
The clutching function correspondence gives you one way of constructing an equivalence $G\simeq \Omega BG$, and you are asking whether that equivalence is the same as the equivalence $G\simeq \Omega BG$. But this is impossible to answer without specifying how the latter equivalence is constructed. 
Aug 24 
comment 
Continuous relations?
@JoonasIlmavirta: That definition will give you closed relations, which are not very wellbehaved other than being symmetric in $X$ and $Y$. They are not closed under composition and include functions that are not continuous as functions. If you replace $K$ by $Y$ (so you are looking locally only on $X$), you get my definition of continuity (since it is local on the domain). 
Aug 24 
awarded  gn.generaltopology 
Aug 23 
revised 
Continuous relations?
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Aug 23 
revised 
Continuous relations?
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Aug 23 
revised 
Continuous relations?
previous edit was incorrect 
Aug 23 
revised 
Continuous relations?
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Aug 23 
revised 
Continuous relations?
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Aug 23 
revised 
Continuous relations?
added 47 characters in body 
Aug 23 
answered  Continuous relations? 
Aug 22 
revised 
Continuous relations?
clarify what kind of "relation" is meant 
Aug 22 
comment 
Continuous relations?
The obvious definition would be to consider relations that are closed as a subset of $X\times Y$, though this fails to generalize continuous functions unless $Y$ is compact Hausdorff. More generally, "relations" can be defined in any category with finite products as subobjects of the product $X\times Y$; this coincides with closed relations in the category of compact Hausdorff spaces (but means "arbitrary relation together with a topology finer than the product topology" in the category of all topological spaces). 
Aug 22 
comment 
Continuous relations?
What does "continuous relations" mean? 