# Subspace vs Subset: Inheritance of Topology

• tomboi03
In summary, the conversation discusses the proof that if Y is a subspace of X and A is a subset of Y, then the topology A inherits as a subspace of Y is the same as the topology it inherits as a subspace of X. This is shown by defining the topologies T_Y and T_X and proving that T_Y is a subset of T_X.
tomboi03
Hey guys...

I'm not sure how I'm suppose to show that if Y is a subspace of X, and A is a subset of Y, then the topology A inherits as a subspace of Y is the same as the topology it inherits as a subspace of X.

I know that a subspace is... Ty = {Y$$\cap$$U| U $$\in$$T}
meaning that its open sets consist of all intersections of open sets of X with Y.
and that a subset is every element of A is also an element of B.

pretty much right? so how do i express this in terms of subset and subspace?

Let TY denote the topology inherited from Y, and TX the topology inherited from X, i.e.
$$T_Y = \{ U \cap A | U \text{ is open in } Y \}$$
and
$$T_X = \{ V \cap A | V \text{ is open in } X \}$$

First let's show that $T_Y \subseteq T_X$. Let $U \in T_Y$ be an open set in the Y-induced topology on A. That means there is some open set U' in Y, such that $U = A \cap U'$. Can you find a set U'' which is open in X, such that $U = A \cap U''$? Because that would show that
$$U \in T_Y \implies U \in T_X$$
and therefore
$$T_Y \subseteq T_X$$.

## What is the difference between subspace and subset?

Subspace refers to a subset of a topological space that inherits its topology from the larger space. A subset, on the other hand, is simply a collection of elements from a larger set without any additional structure or topology.

## How is topology inherited in a subspace?

In a subspace, the topology is inherited by considering the open sets of the larger space and their intersections with the subspace. The resulting collection of open sets forms the topology of the subspace.

## Can a subset have a different topology from its parent space?

Yes, a subset can have a different topology from its parent space. This is because a subset is not required to inherit the topology of the larger space. It can have its own topology, which may or may not be related to the topology of the parent space.

## What is the significance of subspace topology?

The subspace topology is important in understanding the properties of a subset. It allows us to define continuity, convergence, and other topological concepts on a subset, even if they are not defined on the larger space.

## Are there any limitations to using subspace topology?

One limitation of subspace topology is that it only works for topological spaces. If a space has a different type of structure, such as a metric or uniform structure, then the subspace may not inherit it. Additionally, certain properties of the larger space may not hold in the subspace. For example, a compact space may not necessarily have a compact subspace.

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