nLab weak topology

Induced topologies

For the strong topology in functional analysis, see at strong operator topology.

Induced topologies

Definitions

Weak/coarse/initial topology

Suppose

1. $S$ is a set,

2. $\{ (X_i, T_i) \}_{i \in I}$ is a family of topological spaces

3. $\big\{ f_i \colon S \to X_i \big\}_{i \in I}$ an indexed set of functions from $S$ to the family $\{ X_i \}_{i \in I}$.

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a continuous map for every $i \in I$. Then the intersection $\bigcap_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the coarsest/weakest topology $\tau_0$ on $X$ such that each function $f_i\colon S \to X_i$ is a continuous map.

We call $\tau_0$ the weak/coarse/initial topology induced on $S$ by the family of mappings $\{ f_i \}_{i \in I}$. Note that all terms ‘weak topology’, ‘initial topology’, and ‘induced topology’ are used. The subspace topology is a special case, where $I$ is a singleton and the unique function $f_i$ is an injection.

Strong/fine/final topology

Dually, suppose

1. $S$ is a set,

2. $\{ (X_i, T_i) \}_{i \in I}$ a family of topological spaces

3. $\big\{ f_i \colon X_i \to S \big\}_{i \in I}$ a family of functions to $S$ from the family $\{ X_i \}_{i \in I}$.

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a continuous map for every $i \in I$. Then the union $\bigcup_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the finest/strongest topology $\tau_0$ on $S$ such that each function $f_i\colon X_i \to S$ is a continuous map.

We call $\tau_0$ the strong/fine/final topology induced on $S$ by the family of mappings $\{ f_i \}_{i \in I}$. Note that all terms ‘strong topology’, ‘final topology’, and ‘induced topology’ are used. The quotient topology is a special case, where $I$ is a singleton and the unique function $f_i$ is a surjection.

Generalisations

We can perform the first construction in any topological concrete category, where it is a special case of an initial structure for a source or cosink.

We can also perform the second construction in any topological concrete category, where it is a special case of an final structure for a sink.

In functional analysis

In functional analysis, the term ‘weak topology’ is used in a special way. If $V$ is a topological vector space over the ground field $K$, then we may consider the continuous linear functionals on $V$, that is the continuous linear maps from $V$ to $K$. Taking $V$ to be the set $X$ in the general definition above, taking each $T_i$ to be $K$, and taking the continuous linear functionals on $V$ to comprise the family of functions, then we get the weak topology on $V$.

The weak-star topology on the dual space $V^*$ of continuous linear functionals on $V$ is precisely the weak topology induced by the dual (evaluation) functionals on $V^*$

$\left\{V^* \overset{\operatorname{ev}_v}{\to} K, \text{ by } f \mapsto f(v)\right\}_{v \in V}.$

For the strong topology in functional analysis, see the strong operator topology.

References

The original version of this article was posted by Vishal Lama at induced topology.