Induced topologies

Definitions

Suppose $S$ is a set, $\{(X_i,T_i){\}}_{i\in I}$ a family of topological spaces and $\{f_i{\}}_{i\in I}$ a family of functions from $S$ to the family $\{X_i{\}}_{i\in I}$. That is, for each index $i\in I$, $f_i:S\to X_i$. Let $\Gamma$ be 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 $f_i:S\to X_i$ 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.

Dually, suppose $S$ is a set, $\{(X_i,T_i){\}}_{i\in I}$ a family of topological spaces and $\{f_i{\}}_{i\in I}$ a family of functions to $S$ from the family $\{X_i{\}}_{i\in I}$. That is, for each index $i\in I$, $f_i:X_i\to S$. Let $\Gamma$ be 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 finest/strongest topology ${\tau }_0$ on $X$ such that each function $f_i:X_i\to S$ 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 sink.

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? is another special case of a weak topology.

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.