topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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
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
continuous metric space valued function on compact metric space is uniformly continuous
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
Analysis Theorems
In topology, the result of a space attachment (sometimes called an attaching space or adjunction space) is a topological space, denoted $X \cup_{f} Y$, which is constructed by “attaching” or “gluing” two topological spaces $X$ and $Y$ along a topological subspace $A \subset X$ by means of a continuous function $f \colon A \to Y$. The function $f$ is then called the attaching map.
More abstractly, space attachments are pushouts along monomorphisms in the category Top of all topological spaces. The formally dual concept is that of fiber spaces or more generally of fiber products of topological spaces.
Let $X,Y \in Top$ be topological spaces, let $A \subset X$ be a topological subspace and let $f \colon A \to Y$ be a continuous function.
Then the attaching space $X \cup_f Y \in Top$ may be realized as the quotient topological space of the disjoint union space $X \sqcup Y$ by the equivalence relation which identifies a point $x \in A \subset X$ with its image $f(x) \in Y$:
More category theoretically, the attaching space is the pushout in the category Top of topological spaces of the subspace inclusion $i \colon A \hookrightarrow X$ along $f$, i.e. the topological space which is universal with the property that it makes the following square commute:
For more on this see at Top – Universal constructions.
examples of universal constructions of topological spaces:
$\phantom{AAAA}$limits | $\phantom{AAAA}$colimits |
---|---|
$\,$ point space$\,$ | $\,$ empty space $\,$ |
$\,$ product topological space $\,$ | $\,$ disjoint union topological space $\,$ |
$\,$ topological subspace $\,$ | $\,$ quotient topological space $\,$ |
$\,$ fiber space $\,$ | $\,$ space attachment $\,$ |
$\,$ mapping cocylinder, mapping cocone $\,$ | $\,$ mapping cylinder, mapping cone, mapping telescope $\,$ |
$\,$ cell complex, CW-complex $\,$ |