A topological position of the set of continuous maps in the set of upper semicontinuous maps

Abstract

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points,Combining this statement with a result in our previous paper,if X is an infinite compact metric space. We also prove that, for a metric space X, (↓USCC(X), ↓CC(X)) ≈ (Σ, c0) if and only if X is non-compact, locally compact, non-discrete and separable.

The topological structure of function space of transitive maps

“…In [3], Dobrowolski et al showed that the space of real-valued continuous functions of a countable non-discrete metric space with the topology of pointwise convergence is homeomorphic to the subspace c 0 = {(x n ) ∈ R ∞ : lim n→∞ x n = 0} of the countable product R ∞ of real lines. In 2005-2017, the third named author of the present paper and his coauthors obtained structural characteristics of spaces of continuous functions ↓ C F (X) from a k-space X to I = [0, 1] with the Fell topology of hypograph, see [19][20][21][22][23][24][25][26][27][28][29]. For example, ↓C F (X) is homeomorphic to c 0 if ↓ C F (X) is metrizable and the set of isolated points in X is not dense.…”

Hypographs of Upper Semi-continuous Maps and Continuous Maps on a Bounded Open Interval

“…Cp(X, L)≈c 0 if X is a countable non-discrete metric space and L=R or L=I, where Cp(X, L) denotes C(X, L) endowed with the pointwise convergence topology, and c 0 ={(x n ) ∞ − ∈ ) 1 , 1 ( : lim n→∞ x n =0}. In [4] to [7], C(X, I) was endowed with another topology. To introduce this topology, it is necessary to recall the knowledge about hyperspace.…”

“…Theorem 4 [7]. Let X be a compact metric space, |X| denotes the cardinal number of X, and Σ={(x n )∈Q: sup{x n : n∈N + } < 1} be a subspace of Q, then…”

Topological classification of function spaces with the Fell topology II

“…In [14][15][16][17][18][19], the authors gave the topological classification for all metrizable function spaces ↓C F (X) under the condition that X is metrizable. That is, Theorem 1.…”