see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
Suppose
$S$ is a set,
$\{ (X_i, T_i) \}_{i \in I}$ is a family of topological spaces
$\{ f_i \}_{i \in I}$ a family of functions from $S$ to the family $\{ X_i \}_{i \in I}$.
That is, for each index $i \in I$, $f_i\colon S \to X_i$. 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 $f_i\colon S \to X_i$ 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.
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 $i \in I$, $f_i\colon X_i \to S$. 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 $S$ 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.
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, 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 $V^*$
For the strong topology in functional analysis, see the strong operator topology.
The original version of this article was posted by Vishal Lama at induced topology.
See also