On embedding separable spaces C ( L ) in arbitrary spaces C ( K )

Area of Science:

• Topology
• Functional Analysis
• Set Theory

Background:

• Classical results exist for embeddings between Banach spaces of continuous functions.
• Characterizing embeddings between C(L) and C(K) is crucial for understanding their structural relationships.

Purpose of the Study:

• To provide characterizations for isometric and isomorphic embeddings of C(L) into C(K).
• To refine classical theorems by Holsztyński and Gordon under separability conditions.
• To describe the relative cellularities of topological structures within compact spaces.

Main Methods:

• Utilizing concepts from topology and functional analysis.
• Analyzing properties of compact spaces K and L.
• Investigating Banach spaces C(K) and C(L) of continuous real-valued functions.

Main Results:

• Several characterizations for the existence of isometric and isomorphic embeddings of C(L) into C(K) are established.
• For separable C(L), classical theorems by Holsztyński and Gordon are shown to be equivalences.
• New results describe the relative cellularities of the perfect kernel and Cantor-Bendixson derived sets.

Conclusions:

• The study expands and supplements existing knowledge on embeddings in functional analysis.
• Separability plays a key role in strengthening classical embedding theorems.
• The findings offer new insights into the topological properties of compact spaces through their function spaces.