A pair of spaces of upper semi-continuous maps and continuous maps

Topology and its Applications

2020

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Let C(I) be the set of all continuous self-maps from I = [0, 1] with the topology of uniformly convergence. A map f ∈ C(I) is called a transitive map if for every pair of non-empty open sets U, V in I, there exists a positive integer n such that U ∩ f −n (V ) = ∅. We note T (I) and T (I) to be the sets of all transitive maps and its closure in the space C(I). In this paper, we show that T (I) and T (I) are homeomorphic to the separable Hilbert space 2 .

Section: Introduction and Main Resultsmentioning

confidence: 99%

The topological structure of function space of transitive maps

Topology and its Applications

2020

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“…Define ↓H : I →↓V by H(t) = (1 − t)g 1 + tg, then ↓H is well defined and a path from ↓g 1 to ↓g (cf. Proof of Lemma 6 in [7]). Similarly, there exists a path joining ↓g and ↓g 2 in ↓V .…”

Section: Lemma 5 ↓F N Is An Ar and ↓C N (X) Is Homotopy Dense Inmentioning

confidence: 98%

“…In Section 4, we shall also use Theorem C to prove that some spaces are homeomorphic to Q. For more applications of Theorem C, please refer to [2, Chapter 8] and [7]. These applications illustrate the power of the above characterization theorem.…”

Section: 4]) a Space X Is Homeomorphic To The Hilbert Cube Q If Andmentioning

confidence: 99%

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

Sci. China Ser. A-Math.

2009

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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.

“…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.…”

Section: Introduction and The Main Theoremmentioning

confidence: 99%

Topological classification of function spaces with the Fell topology II

Topology and its Applications

Zheng

Chen

2015

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For a Tychonoff space X, let ↓C F (X) denote the collection of the hypographs of all continuous maps from X to [0, 1] with the Fell topology. We show that for a Tychonoff k-space X, ↓C F (X) is homeomorphic to c 0 if and only if ↓C F (X) is metrizable and not Baire if and only if X is a weakly locally compact and hemicompact ℵ 0 -space without dense set of isolated points, where Q = [−1, 1] ω is the Hilbert cube, Σ = {(x n ) ∈ Q : sup |x n | < 1} and c 0 = {(x n ) ∈ Σ : lim x n = 0} are the subspaces of it.