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
Let $(X,\tau)$ be a topological space, and let $C \subset X$ be a closed subset, regarded as a topological subspace $(C,\tau_{sub})$. Then a subset $S \subset C$ is a closed subset of $(C,\tau_{sub})$ precisely if it is closed as a subset of $(X,\tau)$.
If $S \subset C$ is closed in $(C,\tau_{sub})$ this means equivalently that there is an open open subset $V \subset C$ in $(C, \tau_{sub})$ such that
But by the definition of the subspace topology, this means equivalently that there is a subset $U \subset X$ which is open in $(X,\tau)$ such that $V = U \cap C$. Hence the above is equivalent to the existence of an open subset $U \subset X$ such that
But now the condition that $C$ itself is a closed subset of $(X,\tau)$ means equivalently that there is an open subset $W \subset X$ with $C = X \backslash W$. Hence the above is equivalent to the existence of two open subsets $W,U \subset X$ such that
Since the union $W \cup U$ is again open, this implies that $S$ is closed in $(X,\tau)$.
Conversely, that $S \subset X$ is closed in $(X,\tau)$ means that there exists an open $T \subset X$ with $S = X \backslash T \subset X$. This means that $S = S \cap C = (X \backslash T) \cap C = C\backslash T = C \backslash (T \cap C)$, and since $T \cap C$ is open in $(C,\tau_{sub})$ by definition of the subspace topology, this means that $S \subset C$ is closed in $(C, \tau_{sub})$.
Created on May 15, 2017 at 12:15:04. See the history of this page for a list of all contributions to it.