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subspace topology (Definition)

Let $ X$ be a topological space, and let $ Y \subset X$ be a subset. The subspace topology on $ Y$ is the topology whose open sets are those subsets of $ Y$ which equal $ U \cap Y$ for some open set $ U \subset X$.

In this context, the topological space $ Y$ obtained by taking the subspace topology is called a topological subspace, or simply subspace, of $ X$.



"subspace topology" is owned by djao.
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Other names:  relative topology
Also defines:  topological subspace, subspace

Attachments:
subspace topology in a metric space (Theorem) by matte
subspace of a subspace (Theorem) by matte
closed set in a subspace (Theorem) by yark
characterization of subspace topology (Theorem) by mps
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Cross-references: open sets, subset, topological space
There are 142 references to this entry.

This is version 3 of subspace topology, born on 2001-10-25, modified 2003-03-13.
Object id is 499, canonical name is SubspaceTopology.
Accessed 15139 times total.

Classification:
AMS MSC: 54B05 (General topology :: Basic constructions :: Subspaces)
Pending Errata and Addenda
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