We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive ...

For a function f:[0,1]→R, we consider the set E(f) of points at which f cuts real axis. Given f:[0,1]→R and Cantor D⊂[0,1] with {0,1}⊂D, obtain conditions equivalent to conjunction f∈C[0,1] (or f∈C∞[0,1]) D⊂E(f). This generalizes some ideas Zabeti. We observe that, if is continuous, then closed nowhere dense subset f−1[{0}]. Additionally, Intf−1[{0}]=0̸, each x∈{0,1}∩E(f) an accumulation point E...

Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak si...

Given f : X → R ∪ {+∞} a convex and lower semi-continuous function defined on a reflexive Banach space X, and L, a closed linear manifold of X over which f takes at least a real value, the aim of this note is to prove the following Baire category result: in the Euclidean setting, the set of affine functions dominated by f on L for which there is no dominated extension to X is always of first Ba...

These notes will be rather informal in many places. For more precision, refer to Lectures on Polytopes by Günter Ziegler [Zie95], Convex Polytopes by Branko Grünbaum [Grü67], An Introduction to Convex Polytopes by Arne Brøndsted [Brø83], and Convex Polytopes and the Upper Bound Conjecture by Peter McMullen and Geoffrey Shephard [MS71]. See also the handbook papers [BL93, KK95] We will be studyi...

We construct some non-essential H-closed epireflections of W, the category of archimedean l-groups with weak order unit, that are not comparable with any known H-closed epireflections of W other than the divisible hull and the epicompletion. In preparation, we show that the free objects in any Hclosed epireflective subcategory of W must be closed under composition (see section 2 for a precise d...

There are several sections. The goal is to link complex analysis and imbeddings of graphs so that graph imbeddings can be performed systematically using identification spaces based on group actions in the complex plane. We give a short primer of basic calculus and then explain some of the finer aspects that obfuscate a complete solution to the Riemann Hypothesis. This is not central to our main...

Let (X, ) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Assume that X satisfies; if a non-decreasing sequence xn → x in X , then xn x, for all n. Let F be a set valued mapping from X into X with nonempty closed bounded values satisfying; (i) there exists κ ∈ (0, 1) with D(F (x), F (y)) ≤ κd(x, y), for all x y, (ii) if d(x, y) < ε < 1 for some y ∈...

In this paper, we introduce λκ−closed sets and study its properties in generalized topological spaces.