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We now consider some ways of getting new topologies from old ones.
Definition
If A is a subset of a topological space (X, JX), we define the subspace topology JA on A by:
B ∈ JA if B = A ∩ C for some C ∈ JX .
Examples
Restricting the metric on a metric space to a subset gives this topology.
For example, On X = [0, 1] with the usual metric inherited from R, the open sets are the intersection of [0, 1] with open sets of R.
So, for instance, [1, 1/4) = (-1, 1/4) ∩ [0, 1] and so is an open subset of the subspace X.
Remark
Note that as in this example, sets which are open in the subspace are not necessarily open in the "big space".- The subspace topology on Z ⊂ R (with its usual topology/metric) is the discrete topology.
- The subspace topology on the x-axis as a subset of R2 (with its usual topology) is the usual topology on R.
Remark
If we take the inclusion map i: A → X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
Proof
If B ⊂ X is open then i-1(B) = A ∩ B.
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FAQs on Subspace Topology - Topology, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the subspace topology? |  |
Ans. The subspace topology is a topology that is induced on a subset of a topological space. It is a way to define a topology on a subset by considering only the open sets of the larger space that intersect with the subset.
| 2. How is the subspace topology defined? | |
Ans. The subspace topology on a subset A of a topological space X is defined by taking the collection of all sets of the form A ∩ U, where U is an open set in X. These sets form a basis for the subspace topology on A.
| 3. What are the properties of the subspace topology? | |
Ans. The subspace topology inherits certain properties from the larger topological space. For example, if X is Hausdorff, then any subspace of X is also Hausdorff. Similarly, if X is compact, then any subspace of X is also compact. However, the subspace topology may not be connected or path-connected even if the larger space is.
| 4. How is the subspace topology related to continuity? | |
Ans. A function between two topological spaces is continuous if and only if the inverse image of every open set in the codomain is open in the domain. In the case of a subspace, a function is continuous with respect to the subspace topology if and only if its restriction to the subspace is continuous in the larger space.
| 5. Can a subspace have a different topology than the larger space? | |
Ans. Yes, a subspace can have a different topology than the larger space. The subspace topology is a way to define a topology on a subset, and it is not necessarily the same as the topology of the larger space. The subspace topology only considers the open sets of the larger space that intersect with the subset, so it may have fewer open sets than the larger space.
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