In this chapter, we study the cardinal functions on the space C(X) equipped with the uniform, fine and graph topologies. We are primarily interested in five cardinal functions which correspond to the well-known countability properties.

In this chapter, we study the connectedness and some related algebraic properties of the uniform, fine and graph topologies on the space C(X, Y), the set of all continuous functions from a Tychonoff space X to a normed linear space $$(Y,||\cdot ||)$$ . We show that these function spaces are in general not connected and in that case we determine the components and path components of these spaces.

In this chapter, we do a brief study of the compact subsets of C(X, Y) with respect to the uniform, fine and graph topologies, where Y is a metric space and prove the Stone-Weierstrass approximation theorem in detail.

In this chapter, we study some topological properties of the space H(X), the set of all homeomorphisms from a metric space X onto itself, where H(X) has either the uniform topology or the fine topology. In particular, we study the countability and connectedness of the space H(X) with the uniform and fine topologies. Also for the case that $$X=\mathbb {R}^n$$ , three different natural compatible metrics are used to generate three different uniform topologies on $$H(\mathbb {R}^n)$$ . These three homeomorphism spaces are shown to be not homeomorphic to each other for $$n>1$$ , and are also compared to $$H(\mathbb {R}^n)$$ with the fine, point-open and compact-open topologies.

In this chapter, we study various topological properties of the uniform, fine and graph topologies on the space C(X, Y), the set of all continuous functions from a Tychonoff space X to a metric space Y. In particular, we study the metrizability, first countability and various completeness properties of the uniform, fine and graph topologies on C(X, Y).

For the set C(X) of real-valued continuous functions on a Tychonoff space X, the compact-open topology on C(X) is a “set-open topology”. This paper studies the separation and countability properties of the space C(X) having the topology given by the join of the compact-open topology and an “open-set topology” called the open-point topology, that was introduced in Jindal et al. (Topol. Appl. 187: 62–74, 2015).

This paper studies various completeness properties of the open-point and bi-point-open topologies on the space C(X) of all real-valued continuous functions on a Tychonoff space X. The properties range from complete metrizability to the Baire space property.

In the definition of a set-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, we use a certain family of subsets of X and open subsets of R. But instead of using this traditional way to define topologies on C(X), in this paper, we adopt a different approach to define two interesting topologies on C(X). We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.

In [9], two new kinds of topologies called the open-point topology and the bi-point-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, have been introduced and their separation and countability properties have been studied. In the present paper, we study the submetrizability and cardinal functions such as extent, cellularity, weight, pseudocharacter, character and tightness of the spaces C(X) equipped with the open-point and bi-point-open topologies.

A study is made of the countability and connectedness properties of the space H(X) of self-homeomorphisms from a metric space X onto itself, where H(X) has either the uniform topology or the fine topology. Also for the case that X = R-n, three different natural compatible metrics are used to generate three different uniform topologies on H(R-n). These three homeomorphism spaces are shown to be not homeomorphic to each other for n > 1, and are also compared to H(R-n) with the fine, point-open and compact-open topologies.

This study looks at some subgroups of the group H(C(X)) of homeomorphisms on the space C(X) of continuous real-valued functions on a topological space X, where C(X) has the compact-open topology. The main result shows that, for certain spaces X, the subgroup of H(C(X)) generated by the algebraic and vertical homeomorphisms on C(X) is dense in H(C(X)) with the pointwise topology. Also, for X equal to the unit interval, a subgroup of H(C(X)) is developed using integration of the members of C(X), and this subgroup is used as an example and to illustrate certain properties that subgroups of H(C(X)) can have.

The path components and connected components are determined for the space H(C) of homeomorphisms on the complex plane C for the three cases that H(C) has the pointwise topology, the compact-open topology, and the fine topology. The space H(C) is also considered with the uniform topology, but the characterization of the path components and connected components there is left as an open question.

A study is made of two large subgroups, algebraic and bimonotone, of the group of homeomorphisms on C(X), where C(X) is the space of continuous real-valued functions on X with the compact-open topology. The bimonotone subgroup is generated by the union of the subgroup of horizontal homeomorphisms on C(X) and the subgroup of vertical homeomorphisms on C(X). These subgroups are shown to intersect only at the identity. When X is compact, the horizontal and vertical subgroups are shown to be closed subgroups of the group of homeomorphisms on C(X) with the uniform topology, and hence closed in this group with the fine topology; which is known to be a topological group. A number of examples are given, and questions are raised throughout.

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.

When the space C(X) of continuous real-valued functions on X has the uniform topology, the space H(C(X)) of homeomorphisms on C(X) is a topological group when it has the fine topology. This article shows that for certain subgroups F and G of H(C(X)) and H(C(Y)), respectively, there is a natural one-to-one correspondence between a certain set of topological isomorphisms from F onto G and a certain set of homeomorphisms from C(X) onto C(Y) that relate to F and G. A number of examples are given of types of subgroups of H(C(X)) that satisfy this Correspondence Theorem and a weaker version of it.

The space H-f(Y) of homeomorphisms on metric space Y under the fine topology is shown to be a topological group. The space H-f(+)(R) of increasing homeomorphisms on has topological properties much like those of the box product square R-omega, but these two spaces are actually not homeomorphic. Under the motivation of finding a product space that is homeomorphic to H-f(+)(R), the semi-box product R-omega is introduced, and its topological properties are studied. Among other relationships between the two spaces H-f(+) (R) and R-omega is the one that each can be embedded in the other.

We introduce a lower semicontinuous analog, L −(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L −(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L −(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.

The space Hf (Y ) of homeomorphisms on metric space Y under the fine topology is shown to be a topological group. The space H f (R) of increasing homeomorphisms on R has topological properties much like those of the box product □R, but these two spaces are actually not homeomorphic. Under the motivation of finding a product space that is homeomorphic to H f (R), the semi-box product ⊐ R is introduced, and its topological properties are studied. Among other relationships between the two spaces H f (R) and ⊐ R, is the one that each can be embedded in the other.

A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F ( x ) is a nonempty, compact and connected subset of Y for each x ∈ X . We denote by L ( X ), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L ( X ) endowed with the Vietoris topology.